Filters in topology

Short description: Use of filters to describe and characterize all basic topological notions and results.

The power set lattice of the set [math]\displaystyle{ X := \{1, 2, 3, 4\}, }[/math] with the upper set [math]\displaystyle{ \{1, 4\}^{\uparrow X} }[/math] colored dark green. It is a filter, and even a principal filter. It is not an ultrafilter, as it can be extended to the larger nontrivial filter [math]\displaystyle{ \{1\}^{\uparrow X} }[/math] by including also the light green elements. Because [math]\displaystyle{ \{1\}^{\uparrow X} }[/math] cannot be extended any further, it is an ultrafilter.

Filters in topology, a subfield of mathematics, can be used to study topological spaces and define all basic topological notions such as convergence, continuity, compactness, and more. Filters, which are special families of subsets of some given set, also provide a common framework for defining various types of limits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters called ultrafilters have many useful technical properties and they may often be used in place of arbitrary filters.

Filters have generalizations called prefilters (also known as filter bases) and filter subbases, all of which appear naturally and repeatedly throughout topology. Examples include neighborhood filters/bases/subbases and uniformities. Every filter is a prefilter and both are filter subbases. Every prefilter and filter subbase is contained in a unique smallest filter, which they are said to generate. This establishes a relationship between filters and prefilters that may often be exploited to allow one to use whichever of these two notions is more technically convenient. There is a certain preorder on families of sets, denoted by [math]\displaystyle{ \,\leq,\, }[/math] that helps to determine exactly when and how one notion (filter, prefilter, etc.) can or cannot be used in place of another. This preorder's importance is amplified by the fact that it also defines the notion of filter convergence, where by definition, a filter (or prefilter) [math]\displaystyle{ \mathcal{B} }[/math] converges to a point if and only if [math]\displaystyle{ \mathcal{N} \leq \mathcal{B}, }[/math] where [math]\displaystyle{ \mathcal{N} }[/math] is that point's neighborhood filter. Consequently, subordination also plays an important role in many concepts that are related to convergence, such as cluster points and limits of functions. In addition, the relation [math]\displaystyle{ \mathcal{S} \geq \mathcal{B}, }[/math] which denotes [math]\displaystyle{ \mathcal{B} \leq \mathcal{S} }[/math] and is expressed by saying that [math]\displaystyle{ \mathcal{S} }[/math] is subordinate to [math]\displaystyle{ \mathcal{B}, }[/math] also establishes a relationship in which [math]\displaystyle{ \mathcal{S} }[/math] is to [math]\displaystyle{ \mathcal{B} }[/math] as a subsequence is to a sequence (that is, the relation [math]\displaystyle{ \geq, }[/math] which is called subordination, is for filters the analog of "is a subsequence of").

Filters were introduced by Henri Cartan in 1937 and subsequently used by Bourbaki in their book Topologie Générale as an alternative to the similar notion of a net developed in 1922 by E. H. Moore and H. L. Smith. Filters can also be used to characterize the notions of sequence and net convergence. But unlike sequence and net convergence, filter convergence is defined entirely in terms of subsets of the topological space [math]\displaystyle{ X }[/math] and so it provides a notion of convergence that is completely intrinsic to the topological space; indeed, the category of topological spaces can be equivalently defined entirely in terms of filters. Every net induces a canonical filter and dually, every filter induces a canonical net, where this induced net (resp. induced filter) converges to a point if and only if the same is true of the original filter (resp. net). This characterization also holds for many other definitions such as cluster points. These relationships make it possible to switch between filters and nets, and they often also allow one to choose whichever of these two notions (filter or net) is more convenient for the problem at hand. However, assuming that "subnet" is defined using either of its most popular definitions (which are those given by Willard and by Kelley), then in general, this relationship does not extend to subordinate filters and subnets because as detailed below, there exist subordinate filters whose filter/subordinate–filter relationship cannot be described in terms of the corresponding net/subnet relationship; this issue can however be resolved by using a less commonly encountered definition of "subnet", which is that of an AA–subnet.

Thus filters/prefilters and this single preorder [math]\displaystyle{ \,\leq\, }[/math] provide a framework that seamlessly ties together fundamental topological concepts such as topological spaces (via neighborhood filters), neighborhood bases, convergence, various limits of functions, continuity, compactness, sequences (via sequential filters), the filter equivalent of "subsequence" (subordination), uniform spaces, and more; concepts that otherwise seem relatively disparate and whose relationships are less clear.

Motivation

Archetypical example of a filter

The archetypical example of a filter is the neighborhood filter [math]\displaystyle{ \mathcal{N}(x) }[/math] at a point [math]\displaystyle{ x }[/math] in a topological space [math]\displaystyle{ (X, \tau), }[/math] which is the family of sets consisting of all neighborhoods of [math]\displaystyle{ x. }[/math] By definition, a neighborhood of some given point [math]\displaystyle{ x }[/math] is any subset [math]\displaystyle{ B \subseteq X }[/math] whose topological interior contains this point; that is, such that [math]\displaystyle{ x \in \operatorname{Int}_X B. }[/math] Importantly, neighborhoods are not required to be open sets; those are called open neighborhoods. Listed below are those fundamental properties of neighborhood filters that ultimately became the definition of a "filter." A filter on [math]\displaystyle{ X }[/math] is a set [math]\displaystyle{ \mathcal{B} }[/math] of subsets of [math]\displaystyle{ X }[/math] that satisfies all of the following conditions:

1. Not empty:   [math]\displaystyle{ X \in \mathcal{B} }[/math]  –  just as [math]\displaystyle{ X \in \mathcal{N}(x), }[/math] since [math]\displaystyle{ X }[/math] is always a neighborhood of [math]\displaystyle{ x }[/math] (and of anything else that it contains);
2. Does not contain the empty set:   [math]\displaystyle{ \varnothing \not\in \mathcal{B} }[/math]  –  just as no neighborhood of [math]\displaystyle{ x }[/math] is empty;
3. Closed under finite intersections:   If [math]\displaystyle{ B, C \in \mathcal{B} \text{ then } B \cap C \in \mathcal{B} }[/math]  –  just as the intersection of any two neighborhoods of [math]\displaystyle{ x }[/math] is again a neighborhood of [math]\displaystyle{ x }[/math];
4. Upward closed:   If [math]\displaystyle{ B \in \mathcal{B} \text{ and } B \subseteq S \subseteq X }[/math] then [math]\displaystyle{ S \in \mathcal{B} }[/math]  –  just as any subset of [math]\displaystyle{ X }[/math] that contains a neighborhood of [math]\displaystyle{ x }[/math] will necessarily be a neighborhood of [math]\displaystyle{ x }[/math] (this follows from [math]\displaystyle{ \operatorname{Int}_X B \subseteq \operatorname{Int}_X S }[/math] and the definition of "a neighborhood of [math]\displaystyle{ x }[/math]").

Generalizing sequence convergence by using sets − determining sequence convergence without the sequence

A sequence in [math]\displaystyle{ X }[/math] is by definition a map [math]\displaystyle{ \N \to X }[/math] from the natural numbers into the space [math]\displaystyle{ X. }[/math] The original notion of convergence in a topological space was that of a sequence converging to some given point in a space, such as a metric space. With metrizable spaces (or more generally first–countable spaces or Fréchet–Urysohn spaces), sequences usually suffices to characterize, or "describe", most topological properties, such as the closures of subsets or continuity of functions. But there are many spaces where sequences can not be used to describe even basic topological properties like closure or continuity. This failure of sequences was the motivation for defining notions such as nets and filters, which never fail to characterize topological properties.

Nets directly generalize the notion of a sequence since nets are, by definition, maps [math]\displaystyle{ I \to X }[/math] from an arbitrary directed set [math]\displaystyle{ (I, \leq) }[/math] into the space [math]\displaystyle{ X. }[/math] A sequence is just a net whose domain is [math]\displaystyle{ I = \N }[/math] with the natural ordering. Nets have their own notion of convergence, which is a direct generalization of sequence convergence.

Filters generalize sequence convergence in a different way by considering only the values of a sequence. To see how this is done, consider a sequence [math]\displaystyle{ x_\bull = \left(x_i\right)_{i=1}^\infty \text{ in } X, }[/math] which is by definition just a function [math]\displaystyle{ x_{\bullet} : \N \to X }[/math] whose value at [math]\displaystyle{ i \in \N }[/math] is denoted by [math]\displaystyle{ x_i }[/math] rather than by the usual parentheses notation [math]\displaystyle{ x_\bull(i) }[/math] that is commonly used for arbitrary functions. Knowing only the image (sometimes called "the range") [math]\displaystyle{ \operatorname{Im} x_\bull := \left\{x_i : i \in \N\right\} = \left\{x_1, x_2, \ldots\right\} }[/math] of the sequence is not enough to characterize its convergence; multiple sets are needed. It turns out that the needed sets are the following, which are called the tails of the sequence [math]\displaystyle{ x_\bull }[/math]: [math]\displaystyle{ \begin{alignat}{8} x_{\geq 1} =\; &\{&&x_1, &&x_2, &&x_3, &&x_4, &&\ldots&& \,\} \\[0.3ex] x_{\geq 2} =\; &\{&&x_2, &&x_3, &&x_4, &&x_5, &&\ldots&& \,\} \\[0.3ex] x_{\geq 3} =\; &\{&&x_3, &&x_4, &&x_5, &&x_6, &&\ldots&& \,\} \\[0.3ex] & && && &&\;\,\vdots && && && \\[0.3ex] x_{\geq n} =\; &\{&&x_n, \;\;\,&&x_{n+1}, \;&&x_{n+2}, \;&&x_{n+3},&&\ldots&& \,\} \\[0.3ex] & && && &&\;\,\vdots && && && \\[0.3ex] \end{alignat} }[/math]

These sets completely determine this sequence's convergence (or non–convergence) because given any point, this sequence converges to it if and only if for every neighborhood [math]\displaystyle{ U }[/math] (of this point), there is some integer [math]\displaystyle{ n }[/math] such that [math]\displaystyle{ U }[/math] contains all of the points [math]\displaystyle{ x_n, x_{n+1}, \ldots . }[/math] This can be reworded as:

every neighborhood [math]\displaystyle{ U }[/math] must contain some set of the form [math]\displaystyle{ \{x_n, x_{n+1}, \ldots\} }[/math] as a subset.

Or more briefly: every neighborhood must contain some tail [math]\displaystyle{ x_{\geq n} }[/math] as a subset. It is this characterization that can be used with the above family of tails to determine convergence (or non–convergence) of the sequence [math]\displaystyle{ x_\bull : \N \to X. }[/math] Specifically, with the family of sets [math]\displaystyle{ \{x_{\geq 1}, x_{\geq 2}, \ldots\} }[/math] in hand, the function [math]\displaystyle{ x_\bull : \N \to X }[/math] is no longer needed to determine convergence of this sequence (no matter what topology is placed on [math]\displaystyle{ X }[/math]). By generalizing this observation, the notion of "convergence" can be extended from sequences/functions to families of sets.

The above set of tails of a sequence is in general not a filter but it does "generate" a filter via taking its upward closure (which consists of all supersets of all tails). The same is true of other important families of sets such as any neighborhood basis at a given point, which in general is also not a filter but does generate a filter via its upward closure (in particular, it generates the neighborhood filter at that point). The properties that these families share led to the notion of a filter base, also called a prefilter, which by definition is any family having the minimal properties necessary and sufficient for it to generate a filter via taking its upward closure.

Nets versus filters − advantages and disadvantages

Filters and nets each have their own advantages and drawbacks and there's no reason to use one notion exclusively over the other. Depending on what is being proved, a proof may be made significantly easier by using one of these notions instead of the other. Both filters and nets can be used to completely characterize any given topology. Nets are direct generalizations of sequences and can often be used similarly to sequences, so the learning curve for nets is typically much less steep than that for filters. However, filters, and especially ultrafilters, have many more uses outside of topology, such as in set theory, mathematical logic, model theory (ultraproducts, for example), abstract algebra, combinatorics, dynamics, order theory, generalized convergence spaces, Cauchy spaces, and in the definition and use of hyperreal numbers.

Like sequences, nets are functions and so they have the advantages of functions. For example, like sequences, nets can be "plugged into" other functions, where "plugging in" is just function composition. Theorems related to functions and function composition may then be applied to nets. One example is the universal property of inverse limits, which is defined in terms of composition of functions rather than sets and it is more readily applied to functions like nets than to sets like filters (a prominent example of an inverse limit is the Cartesian product). Filters may be awkward to use in certain situations, such as when switching between a filter on a space [math]\displaystyle{ X }[/math] and a filter on a dense subspace [math]\displaystyle{ S \subseteq X. }[/math]

In contrast to nets, filters (and prefilters) are families of sets and so they have the advantages of sets. For example, if [math]\displaystyle{ f }[/math] is surjective then the image [math]\displaystyle{ f^{-1}(\mathcal{B}) := \left\{f^{-1}(B) ~:~ B \in \mathcal{B}\right\} }[/math] under [math]\displaystyle{ f^{-1} }[/math] of an arbitrary filter or prefilter [math]\displaystyle{ \mathcal{B} }[/math] is both easily defined and guaranteed to be a prefilter on [math]\displaystyle{ f }[/math]'s domain, whereas it is less clear how to pullback (unambiguously/without choice) an arbitrary sequence (or net) [math]\displaystyle{ y_\bull }[/math] so as to obtain a sequence or net in the domain (unless [math]\displaystyle{ f }[/math] is also injective and consequently a bijection, which is a stringent requirement). Similarly, the intersection of any collection of filters is once again a filter whereas it is not clear what this could mean for sequences or nets. Because filters are composed of subsets of the very topological space [math]\displaystyle{ X }[/math] that is under consideration, topological set operations (such as closure or interior) may be applied to the sets that constitute the filter. Taking the closure of all the sets in a filter is sometimes useful in functional analysis for instance. Theorems and results about images or preimages of sets under a function may also be applied to the sets that constitute a filter; an example of such a result might be one of continuity's characterizations in terms of preimages of open/closed sets or in terms of the interior/closure operators. Special types of filters called ultrafilters have many useful properties that can significantly help in proving results. One downside of nets is their dependence on the directed sets that constitute their domains, which in general may be entirely unrelated to the space [math]\displaystyle{ X. }[/math] In fact, the class of nets in a given set [math]\displaystyle{ X }[/math] is too large to even be a set (it is a proper class); this is because nets in [math]\displaystyle{ X }[/math] can have domains of any cardinality. In contrast, the collection of all filters (and of all prefilters) on [math]\displaystyle{ X }[/math] is a set whose cardinality is no larger than that of [math]\displaystyle{ \wp(\wp(X)). }[/math] Similar to a topology on [math]\displaystyle{ X, }[/math] a filter on [math]\displaystyle{ X }[/math] is "intrinsic to [math]\displaystyle{ X }[/math]" in the sense that both structures consist entirely of subsets of [math]\displaystyle{ X }[/math] and neither definition requires any set that cannot be constructed from [math]\displaystyle{ X }[/math] (such as [math]\displaystyle{ \N }[/math] or other directed sets, which sequences and nets require).

Preliminaries, notation, and basic notions

In this article, upper case Roman letters like [math]\displaystyle{ S \text{ and } X }[/math] denote sets (but not families unless indicated otherwise) and [math]\displaystyle{ \wp(X) }[/math] will denote the power set of [math]\displaystyle{ X. }[/math] A subset of a power set is called a family of sets (or simply, a family) where it is over [math]\displaystyle{ X }[/math] if it is a subset of [math]\displaystyle{ \wp(X). }[/math] Families of sets will be denoted by upper case calligraphy letters such as [math]\displaystyle{ \mathcal{B}, \mathcal{C}, \text{ and } \mathcal{F}. }[/math] Whenever these assumptions are needed, then it should be assumed that [math]\displaystyle{ X }[/math] is non–empty and that [math]\displaystyle{ \mathcal{B}, \mathcal{F}, }[/math] etc. are families of sets over [math]\displaystyle{ X. }[/math]

The terms "prefilter" and "filter base" are synonyms and will be used interchangeably.

Warning about competing definitions and notation

There are unfortunately several terms in the theory of filters that are defined differently by different authors. These include some of the most important terms such as "filter." While different definitions of the same term usually have significant overlap, due to the very technical nature of filters (and point–set topology), these differences in definitions nevertheless often have important consequences. When reading mathematical literature, it is recommended that readers check how the terminology related to filters is defined by the author. For this reason, this article will clearly state all definitions as they are used. Unfortunately, not all notation related to filters is well established and some notation varies greatly across the literature (for example, the notation for the set of all prefilters on a set) so in such cases this article uses whatever notation is most self describing or easily remembered.

The theory of filters and prefilters is well developed and has a plethora of definitions and notations, many of which are now unceremoniously listed to prevent this article from becoming prolix and to allow for the easy look up of notation and definitions. Their important properties are described later.

Sets operations

The upward closure or isotonization in [math]\displaystyle{ X }[/math] of a family of sets [math]\displaystyle{ \mathcal{B} \subseteq \wp(X) }[/math] is

[math]\displaystyle{ \mathcal{B}^{\uparrow X} := \{S \subseteq X ~:~ B \subseteq S \text{ for some } B \in \mathcal{B} \,\} = {\textstyle\bigcup\limits_{B \in \mathcal{B}}} \{S ~:~ B \subseteq S \subseteq X\} }[/math]

and similarly the downward closure of [math]\displaystyle{ \mathcal{B} }[/math] is [math]\displaystyle{ \mathcal{B}^{\downarrow} := \{S \subseteq B ~:~ B \in \mathcal{B} \,\} = {\textstyle\bigcup\limits_{B \in \mathcal{B}}} \wp(B). }[/math]

Notation and Definition	Name
[math]\displaystyle{ \ker \mathcal{B} = \bigcap_{B \in \mathcal{B}} B }[/math]	Kernel of [math]\displaystyle{ \mathcal{B} }[/math]
[math]\displaystyle{ S \setminus \mathcal{B} := \{S \setminus B ~:~ B \in \mathcal{B}\} = \{S\} \,(\setminus)\, \mathcal{B} }[/math]	Dual of [math]\displaystyle{ \mathcal{B} \text{ in } S }[/math] where [math]\displaystyle{ S }[/math] is a set.
[math]\displaystyle{ \mathcal{B}\big\vert_S := \{B \cap S ~:~ B \in \mathcal{B}\} = \mathcal{B} \,(\cap)\, \{S\} }[/math]	Trace of [math]\displaystyle{ \mathcal{B} \text{ on } S }[/math] or the restriction of [math]\displaystyle{ \mathcal{B} \text{ to } S }[/math] where [math]\displaystyle{ S }[/math] is a set; sometimes denoted by [math]\displaystyle{ \mathcal{B} \cap S }[/math]
[math]\displaystyle{ \mathcal{B} \,(\cap)\, \mathcal{C} = \{B \cap C ~:~ B \in \mathcal{B} \text{ and } C \in \mathcal{C}\} }[/math]	Elementwise (set) intersection ([math]\displaystyle{ \mathcal{B} \cap \mathcal{C} }[/math] will denote the usual intersection)
[math]\displaystyle{ \mathcal{B} \,(\cup)\, \mathcal{C} = \{B \cup C ~:~ B \in \mathcal{B} \text{ and } C \in \mathcal{C}\} }[/math]	Elementwise (set) union ([math]\displaystyle{ \mathcal{B} \cup \mathcal{C} }[/math] will denote the usual union)
[math]\displaystyle{ \mathcal{B} \,(\setminus)\, \mathcal{C} = \{B \setminus C ~:~ B \in \mathcal{B} \text{ and } C \in \mathcal{C}\} }[/math]	Elementwise (set) subtraction ([math]\displaystyle{ \mathcal{B} \setminus \mathcal{C} }[/math] will denote the usual set subtraction)
[math]\displaystyle{ \wp(X) = \{S ~:~ S \subseteq X\} }[/math]	Power set of a set [math]\displaystyle{ X }[/math]

For any two families [math]\displaystyle{ \mathcal{C} \text{ and } \mathcal{F}, }[/math] declare that [math]\displaystyle{ \mathcal{C} \leq \mathcal{F} }[/math] if and only if for every [math]\displaystyle{ C \in \mathcal{C} }[/math] there exists some [math]\displaystyle{ F \in \mathcal{F} \text{ such that } F \subseteq C, }[/math] in which case it is said that [math]\displaystyle{ \mathcal{C} }[/math] is coarser than [math]\displaystyle{ \mathcal{F} }[/math] and that [math]\displaystyle{ \mathcal{F} }[/math] is finer than (or subordinate to) [math]\displaystyle{ \mathcal{C}. }[/math] The notation [math]\displaystyle{ \mathcal{F} \vdash \mathcal{C} \text{ or } \mathcal{F} \geq \mathcal{C} }[/math] may also be used in place of [math]\displaystyle{ \mathcal{C} \leq \mathcal{F}. }[/math]

If [math]\displaystyle{ \mathcal{C} \leq \mathcal{F} }[/math] and [math]\displaystyle{ \mathcal{F} \leq \mathcal{C} }[/math] then [math]\displaystyle{ \mathcal{C} \text{ and } \mathcal{F} }[/math] are said to be equivalent (with respect to subordination).

Two families [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{C} }[/math] mesh, written [math]\displaystyle{ \mathcal{B} \# \mathcal{C}, }[/math] if [math]\displaystyle{ B \cap C \neq \varnothing \text{ for all } B \in \mathcal{B} \text{ and } C \in \mathcal{C}. }[/math]

Throughout, [math]\displaystyle{ f }[/math] is a map.

Notation and Definition	Name
[math]\displaystyle{ f^{-1}(\mathcal{B}) = \left\{f^{-1}(B) ~:~ B \in \mathcal{B}\right\} }[/math]	Image of [math]\displaystyle{ \mathcal{B} \text{ under } f^{-1}, }[/math] or the preimage of [math]\displaystyle{ \mathcal{B} }[/math] under [math]\displaystyle{ f }[/math]
[math]\displaystyle{ f(\mathcal{B}) = \{f(B) ~:~ B \in \mathcal{B}\} }[/math]	Image of [math]\displaystyle{ \mathcal{B} }[/math] under [math]\displaystyle{ f }[/math]
[math]\displaystyle{ \operatorname{image} f = f(\operatorname{domain} f) }[/math]	Image (or range) of [math]\displaystyle{ f }[/math]

Topology notation

Denote the set of all topologies on a set [math]\displaystyle{ X \text{ by } \operatorname{Top}(X). }[/math] Suppose [math]\displaystyle{ \tau \in \operatorname{Top}(X), }[/math] [math]\displaystyle{ S \subseteq X }[/math] is any subset, and [math]\displaystyle{ x \in X }[/math] is any point.

Notation and Definition	Name
[math]\displaystyle{ \tau(S) = \{O \in \tau ~:~ S \subseteq O\} }[/math]	Set or prefilter of open neighborhoods of [math]\displaystyle{ S \text{ in } (X, \tau) }[/math]
[math]\displaystyle{ \tau(x) = \{O \in \tau ~:~ x \in O\} }[/math]	Set or prefilter of open neighborhoods of [math]\displaystyle{ x \text{ in } (X, \tau) }[/math]
[math]\displaystyle{ \mathcal{N}_{\tau}(S) = \mathcal{N}(S) := \tau(S)^{\uparrow X} }[/math]	Set or filter of neighborhoods of [math]\displaystyle{ S \text{ in } (X, \tau) }[/math]
[math]\displaystyle{ \mathcal{N}_{\tau}(x) = \mathcal{N}(x) := \tau(x)^{\uparrow X} }[/math]	Set or filter of neighborhoods of [math]\displaystyle{ x \text{ in } (X, \tau) }[/math]

If [math]\displaystyle{ \varnothing \neq S \subseteq X }[/math] then [math]\displaystyle{ \tau(S) = {\textstyle\bigcap\limits_{s \in S}} \tau(s) \text{ and } \mathcal{N}_{\tau}(S) = {\textstyle\bigcap\limits_{s \in S}} \mathcal{N}_{\tau}(s). }[/math]

Nets and their tails

A directed set is a set [math]\displaystyle{ I }[/math] together with a preorder, which will be denoted by [math]\displaystyle{ \,\leq\, }[/math] (unless explicitly indicated otherwise), that makes [math]\displaystyle{ (I, \leq) }[/math] into an (upward) directed set; this means that for all [math]\displaystyle{ i, j \in I, }[/math] there exists some [math]\displaystyle{ k \in I }[/math] such that [math]\displaystyle{ i \leq k \text{ and } j \leq k. }[/math] For any indices [math]\displaystyle{ i \text{ and } j, }[/math] the notation [math]\displaystyle{ j \geq i }[/math] is defined to mean [math]\displaystyle{ i \leq j }[/math] while [math]\displaystyle{ i \lt j }[/math] is defined to mean that [math]\displaystyle{ i \leq j }[/math] holds but it is not true that [math]\displaystyle{ j \leq i }[/math] (if [math]\displaystyle{ \,\leq\, }[/math] is antisymmetric then this is equivalent to [math]\displaystyle{ i \leq j \text{ and } i \neq j }[/math]).

A net in [math]\displaystyle{ X }[/math] is a map from a non–empty directed set into [math]\displaystyle{ X. }[/math] The notation [math]\displaystyle{ x_\bull = \left(x_i\right)_{i \in I} }[/math] will be used to denote a net with domain [math]\displaystyle{ I. }[/math]

Notation and Definition	Name
[math]\displaystyle{ I_{\geq i} = \{j \in I ~:~ j \geq i\} }[/math]	Tail or section of [math]\displaystyle{ I }[/math] starting at [math]\displaystyle{ i \in I }[/math] where [math]\displaystyle{ (I, \leq) }[/math] is a directed set.
[math]\displaystyle{ x_{\geq i} = \left\{x_j ~:~ j \geq i \text{ and } j \in I\right\} }[/math]	Tail or section of [math]\displaystyle{ x_\bull = \left(x_i\right)_{i \in I} }[/math] starting at [math]\displaystyle{ i \in I }[/math]
[math]\displaystyle{ \operatorname{Tails}\left(x_\bull\right) = \left\{x_{\geq i} ~:~ i \in I\right\} }[/math]	Set or prefilter of tails/sections of [math]\displaystyle{ x_\bull. }[/math] Also called the eventuality filter base generated by (the tails of) [math]\displaystyle{ x_\bull = \left(x_i\right)_{i \in I}. }[/math] If [math]\displaystyle{ x_\bull }[/math] is a sequence then [math]\displaystyle{ \operatorname{Tails}\left(x_\bull\right) }[/math] is also called the sequential filter base.
[math]\displaystyle{ \operatorname{TailsFilter}\left(x_\bull\right) = \operatorname{Tails}\left(x_\bull\right)^{\uparrow X} }[/math]	(Eventuality) filter of/generated by (tails of) [math]\displaystyle{ x_\bull }[/math]
[math]\displaystyle{ f\left(I_{\geq i}\right) = \{f(j) ~:~ j \geq i \text{ and } j \in I\} }[/math]	Tail or section of a net [math]\displaystyle{ f : I \to X }[/math] starting at [math]\displaystyle{ i \in I }[/math] where [math]\displaystyle{ (I, \leq) }[/math] is a directed set.

Warning about using strict comparison

If [math]\displaystyle{ x_\bull = \left(x_i\right)_{i \in I} }[/math] is a net and [math]\displaystyle{ i \in I }[/math] then it is possible for the set [math]\displaystyle{ x_{\gt i} = \left\{x_j ~:~ j \gt i \text{ and } j \in I\right\}, }[/math] which is called the tail of [math]\displaystyle{ x_\bull }[/math] after [math]\displaystyle{ i }[/math], to be empty (for example, this happens if [math]\displaystyle{ i }[/math] is an upper bound of the directed set [math]\displaystyle{ I }[/math]). In this case, the family [math]\displaystyle{ \left\{x_{\gt i} ~:~ i \in I\right\} }[/math] would contain the empty set, which would prevent it from being a prefilter (defined later). This is the (important) reason for defining [math]\displaystyle{ \operatorname{Tails}\left(x_\bull\right) }[/math] as [math]\displaystyle{ \left\{x_{\geq i} ~:~ i \in I\right\} }[/math] rather than [math]\displaystyle{ \left\{x_{\gt i} ~:~ i \in I\right\} }[/math] or even [math]\displaystyle{ \left\{x_{\gt i} ~:~ i \in I\right\}\cup \left\{x_{\geq i} ~:~ i \in I\right\} }[/math] and it is for this reason that in general, when dealing with the prefilter of tails of a net, the strict inequality [math]\displaystyle{ \,\lt \, }[/math] may not be used interchangeably with the inequality [math]\displaystyle{ \,\leq. }[/math]

Filters and prefilters

The following is a list of properties that a family [math]\displaystyle{ \mathcal{B} }[/math] of sets may possess and they form the defining properties of filters, prefilters, and filter subbases. Whenever it is necessary, it should be assumed that [math]\displaystyle{ \mathcal{B} \subseteq \wp(X). }[/math]

The family of sets [math]\displaystyle{ \mathcal{B} }[/math] is:

1. Proper or nondegenerate if [math]\displaystyle{ \varnothing \not\in \mathcal{B}. }[/math] Otherwise, if [math]\displaystyle{ \varnothing \in \mathcal{B}, }[/math] then it is called improper or degenerate.
2. Directed downward if whenever [math]\displaystyle{ A, B \in \mathcal{B} }[/math] then there exists some [math]\displaystyle{ C \in \mathcal{B} }[/math] such that [math]\displaystyle{ C \subseteq A \cap B. }[/math]
   • This property can be characterized in terms of directedness, which explains the word "directed": A binary relation [math]\displaystyle{ \,\preceq\, }[/math] on [math]\displaystyle{ \mathcal{B} }[/math] is called (upward) directed if for any two [math]\displaystyle{ A \text{ and } B, }[/math] there is some [math]\displaystyle{ C }[/math] satisfying [math]\displaystyle{ A \preceq C \text{ and } B \preceq C. }[/math] Using [math]\displaystyle{ \,\supseteq\, }[/math] in place of [math]\displaystyle{ \,\preceq\, }[/math] gives the definition of directed downward whereas using [math]\displaystyle{ \,\subseteq\, }[/math] instead gives the definition of directed upward. Explicitly, [math]\displaystyle{ \mathcal{B} }[/math] is directed downward (resp. directed upward) if and only if for all [math]\displaystyle{ A, B \in \mathcal{B}, }[/math] there exists some "greater" [math]\displaystyle{ C \in \mathcal{B} }[/math] such that [math]\displaystyle{ A \supseteq C \text{ and } B \supseteq C }[/math] (resp. such that [math]\displaystyle{ A \subseteq C \text{ and } B \subseteq C }[/math]) − where the "greater" element is always on the right hand side, − which can be rewritten as [math]\displaystyle{ A \cap B \supseteq C }[/math] (resp. as [math]\displaystyle{ A \cup B \subseteq C }[/math]).
3. Closed under finite intersections (resp. unions) if the intersection (resp. union) of any two elements of [math]\displaystyle{ \mathcal{B} }[/math] is an element of [math]\displaystyle{ \mathcal{B}. }[/math]
   • If [math]\displaystyle{ \mathcal{B} }[/math] is closed under finite intersections then [math]\displaystyle{ \mathcal{B} }[/math] is necessarily directed downward. The converse is generally false.
4. Upward closed or Isotone in [math]\displaystyle{ X }[/math] if [math]\displaystyle{ \mathcal{B} \subseteq \wp(X) \text{ and } \mathcal{B} = \mathcal{B}^{\uparrow X}, }[/math] or equivalently, if whenever [math]\displaystyle{ B \in \mathcal{B} }[/math] and some set [math]\displaystyle{ C }[/math] satisfies [math]\displaystyle{ B \subseteq C \subseteq X, \text{ then } C \in \mathcal{B}. }[/math] Similarly, [math]\displaystyle{ \mathcal{B} }[/math] is downward closed if [math]\displaystyle{ \mathcal{B} = \mathcal{B}^{\downarrow}. }[/math] An upward (respectively, downward) closed set is also called an upper set or upset (resp. a lower set or down set).
   • The family [math]\displaystyle{ \mathcal{B}^{\uparrow X}, }[/math] which is the upward closure of [math]\displaystyle{ \mathcal{B} \text{ in } X, }[/math] is the unique smallest (with respect to [math]\displaystyle{ \,\subseteq }[/math]) isotone family of sets over [math]\displaystyle{ X }[/math] having [math]\displaystyle{ \mathcal{B} }[/math] as a subset.

Many of the properties of [math]\displaystyle{ \mathcal{B} }[/math] defined above and below, such as "proper" and "directed downward," do not depend on [math]\displaystyle{ X, }[/math] so mentioning the set [math]\displaystyle{ X }[/math] is optional when using such terms. Definitions involving being "upward closed in [math]\displaystyle{ X, }[/math]" such as that of "filter on [math]\displaystyle{ X, }[/math]" do depend on [math]\displaystyle{ X }[/math] so the set [math]\displaystyle{ X }[/math] should be mentioned if it is not clear from context.

A family [math]\displaystyle{ \mathcal{B} }[/math] is/is a(n):

1. Ideal if [math]\displaystyle{ \mathcal{B} \neq \varnothing }[/math] is downward closed and closed under finite unions.
2. Dual ideal on [math]\displaystyle{ X }[/math] if [math]\displaystyle{ \mathcal{B} \neq \varnothing }[/math] is upward closed in [math]\displaystyle{ X }[/math] and also closed under finite intersections. Equivalently, [math]\displaystyle{ \mathcal{B} \neq \varnothing }[/math] is a dual ideal if for all [math]\displaystyle{ R, S \subseteq X, }[/math] [math]\displaystyle{ R \cap S \in \mathcal{B} \;\text{ if and only if }\; R, S \in \mathcal{B}. }[/math]
   • Explanation of the word "dual": A family [math]\displaystyle{ \mathcal{B} }[/math] is a dual ideal (resp. an ideal) on [math]\displaystyle{ X }[/math] if and only if the dual of [math]\displaystyle{ \mathcal{B} \text{ in } X, }[/math] which is the family [math]\displaystyle{ X \setminus \mathcal{B} := \{X \setminus B ~:~ B \in \mathcal{B}\}, }[/math] is an ideal (resp. a dual ideal) on [math]\displaystyle{ X. }[/math] In other words, dual ideal means "dual of an ideal". The dual of the dual is the original family, meaning [math]\displaystyle{ X \setminus (X \setminus \mathcal{B}) = \mathcal{B}. }[/math]
3. Filter on [math]\displaystyle{ X }[/math] if [math]\displaystyle{ \mathcal{B} }[/math] is a proper dual ideal on [math]\displaystyle{ X. }[/math] That is, a filter on [math]\displaystyle{ X }[/math] is a non−empty subset of [math]\displaystyle{ \wp(X) \setminus \{\varnothing\} }[/math] that is closed under finite intersections and upward closed in [math]\displaystyle{ X. }[/math] Equivalently, it is a prefilter that is upward closed in [math]\displaystyle{ X. }[/math] In words, a filter on [math]\displaystyle{ X }[/math] is a family of sets over [math]\displaystyle{ X }[/math] that (1) is not empty (or equivalently, it contains [math]\displaystyle{ X }[/math]), (2) is closed under finite intersections, (3) is upward closed in [math]\displaystyle{ X, }[/math] and (4) does not have the empty set as an element.
   • Warning: Some authors, particularly algebrists, use "filter" to mean a dual ideal; others, particularly topologists, use "filter" to mean a proper/non–degenerate dual ideal. It is recommended that readers always check how "filter" is defined when reading mathematical literature. However, the definitions of "ultrafilter," "prefilter," and "filter subbase" always require non-degeneracy. This article uses Henri Cartan's original definition of "filter", which required non–degeneracy.
   • The power set [math]\displaystyle{ \wp(X) }[/math] is the one and only dual ideal on [math]\displaystyle{ X }[/math] that is not also a filter. Excluding [math]\displaystyle{ \wp(X) }[/math] from the definition of "filter" in topology has the same benefit as excluding [math]\displaystyle{ 1 }[/math] from the definition of "prime number": it obviates the need to specify "non-degenerate" (the analog of "non-unital" or "non-[math]\displaystyle{ 1 }[/math]") in many important results, thereby making their statements less awkward.
4. Prefilter or filter base if [math]\displaystyle{ \mathcal{B} \neq \varnothing }[/math] is proper and directed downward. Equivalently, [math]\displaystyle{ \mathcal{B} }[/math] is called a prefilter if its upward closure [math]\displaystyle{ \mathcal{B}^{\uparrow X} }[/math] is a filter. It can also be defined as any family that is equivalent to some filter. A proper family [math]\displaystyle{ \mathcal{B} \neq \varnothing }[/math] is a prefilter if and only if [math]\displaystyle{ \mathcal{B} \,(\cap)\, \mathcal{B} \leq \mathcal{B}. }[/math] A family is a prefilter if and only if the same is true of its upward closure.
   • If [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter then its upward closure [math]\displaystyle{ \mathcal{B}^{\uparrow X} }[/math] is the unique smallest (relative to [math]\displaystyle{ \subseteq }[/math]) filter on [math]\displaystyle{ X }[/math] containing [math]\displaystyle{ \mathcal{B} }[/math] and it is called the filter generated by [math]\displaystyle{ \mathcal{B}. }[/math] A filter [math]\displaystyle{ \mathcal{F} }[/math] is said to be generated by a prefilter [math]\displaystyle{ \mathcal{B} }[/math] if [math]\displaystyle{ \mathcal{F} = \mathcal{B}^{\uparrow X}, }[/math] in which [math]\displaystyle{ \mathcal{B} }[/math] is called a filter base for [math]\displaystyle{ \mathcal{F}. }[/math]
   • Unlike a filter, a prefilter is not necessarily closed under finite intersections.
5. π–system if [math]\displaystyle{ \mathcal{B} \neq \varnothing }[/math] is closed under finite intersections. Every non–empty family [math]\displaystyle{ \mathcal{B} }[/math] is contained in a unique smallest π–system called the π–system generated by [math]\displaystyle{ \mathcal{B}, }[/math] which is sometimes denoted by [math]\displaystyle{ \pi(\mathcal{B}). }[/math] It is equal to the intersection of all π–systems containing [math]\displaystyle{ \mathcal{B} }[/math] and also to the set of all possible finite intersections of sets from [math]\displaystyle{ \mathcal{B} }[/math]: [math]\displaystyle{ \pi(\mathcal{B}) = \left\{B_1 \cap \cdots \cap B_n ~:~ n \geq 1 \text{ and } B_1, \ldots, B_n \in \mathcal{B}\right\}. }[/math]
   • A π–system is a prefilter if and only if it is proper. Every filter is a proper π–system and every proper π–system is a prefilter but the converses do not hold in general.
   • A prefilter is equivalent to the π–system generated by it and both of these families generate the same filter on [math]\displaystyle{ X. }[/math]
6. Filter subbase and centered if [math]\displaystyle{ \mathcal{B} \neq \varnothing }[/math] and [math]\displaystyle{ \mathcal{B} }[/math] satisfies any of the following equivalent conditions:
   1. [math]\displaystyle{ \mathcal{B} }[/math] has the finite intersection property, which means that the intersection of any finite family of (one or more) sets in [math]\displaystyle{ \mathcal{B} }[/math] is not empty; explicitly, this means that whenever [math]\displaystyle{ n \geq 1 \text{ and } B_1, \ldots, B_n \in \mathcal{B} }[/math] then [math]\displaystyle{ \varnothing \neq B_1 \cap \cdots \cap B_n. }[/math]
   2. The π–system generated by [math]\displaystyle{ \mathcal{B} }[/math] is proper; that is, [math]\displaystyle{ \varnothing \not\in \pi(\mathcal{B}). }[/math]
   3. The π–system generated by [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter.
   4. [math]\displaystyle{ \mathcal{B} }[/math] is a subset of some prefilter.
   5. [math]\displaystyle{ \mathcal{B} }[/math] is a subset of some filter.
   • Assume that [math]\displaystyle{ \mathcal{B} }[/math] is a filter subbase. Then there is a unique smallest (relative to [math]\displaystyle{ \subseteq }[/math]) filter [math]\displaystyle{ \mathcal{F}_{\mathcal{B}} \text{ on } X }[/math] containing [math]\displaystyle{ \mathcal{B} }[/math] called the filter generated by [math]\displaystyle{ \mathcal{B} }[/math], and [math]\displaystyle{ \mathcal{B} }[/math] is said to be a filter subbase for this filter. This filter is equal to the intersection of all filters on [math]\displaystyle{ X }[/math] that are supersets of [math]\displaystyle{ \mathcal{B}. }[/math] The π–system generated by [math]\displaystyle{ \mathcal{B}, }[/math] denoted by [math]\displaystyle{ \pi(\mathcal{B}), }[/math] will be a prefilter and a subset of [math]\displaystyle{ \mathcal{F}_{\mathcal{B}}. }[/math] Moreover, the filter generated by [math]\displaystyle{ \mathcal{B} }[/math] is equal to the upward closure of [math]\displaystyle{ \pi(\mathcal{B}), }[/math] meaning [math]\displaystyle{ \pi(\mathcal{B})^{\uparrow X} = \mathcal{F}_{\mathcal{B}}. }[/math] However, [math]\displaystyle{ \mathcal{B}^{\uparrow X} = \mathcal{F}_{\mathcal{B}} }[/math] if and only if [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter (although [math]\displaystyle{ \mathcal{B}^{\uparrow X} }[/math] is always an upward closed filter subbase for [math]\displaystyle{ \mathcal{F}_{\mathcal{B}} }[/math]).
   • A [math]\displaystyle{ \subseteq }[/math] –smallest (meaning smallest relative to [math]\displaystyle{ \subseteq }[/math] ) prefilter containing a filter subbase [math]\displaystyle{ \mathcal{B} }[/math] will exist only under certain circumstances. It exists, for example, if the filter subbase [math]\displaystyle{ \mathcal{B} }[/math] happens to also be a prefilter. It also exists if the filter (or equivalently, the π–system) generated by [math]\displaystyle{ \mathcal{B} }[/math] is principal, in which case [math]\displaystyle{ \mathcal{B} \cup \{\ker \mathcal{B}\} }[/math] is the unique smallest prefilter containing [math]\displaystyle{ \mathcal{B}. }[/math] Otherwise, in general, a [math]\displaystyle{ \subseteq }[/math] –smallest prefilter containing [math]\displaystyle{ \mathcal{B} }[/math] might not exist. For this reason, some authors may refer to the π–system generated by [math]\displaystyle{ \mathcal{B} }[/math] as the prefilter generated by [math]\displaystyle{ \mathcal{B}. }[/math] However, if a [math]\displaystyle{ \subseteq }[/math] –smallest prefilter does exist (say it is denoted by [math]\displaystyle{ \operatorname{minPre} \mathcal{B} }[/math]) then contrary to usual expectations, it is not necessarily equal to "the prefilter generated by [math]\displaystyle{ \mathcal{B} }[/math]" (that is, [math]\displaystyle{ \operatorname{minPre} \mathcal{B} \neq \pi(\mathcal{B}) }[/math] is possible). And if the filter subbase [math]\displaystyle{ \mathcal{B} }[/math] happens to also be a prefilter but not a π-system then unfortunately, "the prefilter generated by this prefilter" (meaning [math]\displaystyle{ \pi(\mathcal{B}) }[/math]) will not be [math]\displaystyle{ \mathcal{B} = \operatorname{minPre} \mathcal{B} }[/math] (that is, [math]\displaystyle{ \pi(\mathcal{B}) \neq \mathcal{B} }[/math] is possible even when [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter), which is why this article will prefer the accurate and unambiguous terminology of "the π–system generated by [math]\displaystyle{ \mathcal{B} }[/math]".
7. Subfilter of a filter [math]\displaystyle{ \mathcal{F} }[/math] and that [math]\displaystyle{ \mathcal{F} }[/math] is a superfilter of [math]\displaystyle{ \mathcal{B} }[/math] if [math]\displaystyle{ \mathcal{B} }[/math] is a filter and [math]\displaystyle{ \mathcal{B} \subseteq \mathcal{F} }[/math] where for filters, [math]\displaystyle{ \mathcal{B} \subseteq \mathcal{F} \text{ if and only if } \mathcal{B} \leq \mathcal{F}. }[/math]
   • Importantly, the expression "is a superfilter of" is for filters the analog of "is a subsequence of". So despite having the prefix "sub" in common, "is a subfilter of" is actually the reverse of "is a subsequence of." However, [math]\displaystyle{ \mathcal{B} \leq \mathcal{F} }[/math] can also be written [math]\displaystyle{ \mathcal{F} \vdash \mathcal{B} }[/math] which is described by saying "[math]\displaystyle{ \mathcal{F} }[/math] is subordinate to [math]\displaystyle{ \mathcal{B}. }[/math]" With this terminology, "is subordinate to" becomes for filters (and also for prefilters) the analog of "is a subsequence of," which makes this one situation where using the term "subordinate" and symbol [math]\displaystyle{ \,\vdash\, }[/math] may be helpful.

There are no prefilters on [math]\displaystyle{ X = \varnothing }[/math] (nor are there any nets valued in [math]\displaystyle{ \varnothing }[/math]), which is why this article, like most authors, will automatically assume without comment that [math]\displaystyle{ X \neq \varnothing }[/math] whenever this assumption is needed.

Basic examples

Named examples

• The singleton set [math]\displaystyle{ \mathcal{B} = \{X\} }[/math] is called the indiscrete or trivial filter on [math]\displaystyle{ X. }[/math] It is the unique minimal filter on [math]\displaystyle{ X }[/math] because it is a subset of every filter on [math]\displaystyle{ X }[/math]; however, it need not be a subset of every prefilter on [math]\displaystyle{ X. }[/math]
• The dual ideal [math]\displaystyle{ \wp(X) }[/math] is also called the degenerate filter on [math]\displaystyle{ X }[/math] (despite not actually being a filter). It is the only dual ideal on [math]\displaystyle{ X }[/math] that is not a filter on [math]\displaystyle{ X. }[/math]
• If [math]\displaystyle{ (X, \tau) }[/math] is a topological space and [math]\displaystyle{ x \in X, }[/math] then the neighborhood filter [math]\displaystyle{ \mathcal{N}(x) }[/math] at [math]\displaystyle{ x }[/math] is a filter on [math]\displaystyle{ X. }[/math] By definition, a family [math]\displaystyle{ \mathcal{B} \subseteq \wp(X) }[/math] is called a neighborhood basis (resp. a neighborhood subbase) at [math]\displaystyle{ x \text{ for } (X, \tau) }[/math] if and only if [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter (resp. [math]\displaystyle{ \mathcal{B} }[/math] is a filter subbase) and the filter on [math]\displaystyle{ X }[/math] that [math]\displaystyle{ \mathcal{B} }[/math] generates is equal to the neighborhood filter [math]\displaystyle{ \mathcal{N}(x). }[/math] The subfamily [math]\displaystyle{ \tau(x) \subseteq \mathcal{N}(x) }[/math] of open neighborhoods is a filter base for [math]\displaystyle{ \mathcal{N}(x). }[/math] Both prefilters [math]\displaystyle{ \mathcal{N}(x) \text{ and } \tau(x) }[/math] also form a bases for topologies on [math]\displaystyle{ X, }[/math] with the topology generated [math]\displaystyle{ \tau(x) }[/math] being coarser than [math]\displaystyle{ \tau. }[/math] This example immediately generalizes from neighborhoods of points to neighborhoods of non–empty subsets [math]\displaystyle{ S \subseteq X. }[/math]
• [math]\displaystyle{ \mathcal{B} }[/math] is an elementary prefilter if [math]\displaystyle{ \mathcal{B} = \operatorname{Tails}\left(x_\bull\right) }[/math] for some sequence of points [math]\displaystyle{ x_\bull = \left(x_i\right)_{i=1}^\infty. }[/math]
• [math]\displaystyle{ \mathcal{B} }[/math] is an elementary filter or a sequential filter on [math]\displaystyle{ X }[/math] if [math]\displaystyle{ \mathcal{B} }[/math] is a filter on [math]\displaystyle{ X }[/math] generated by some elementary prefilter. The filter of tails generated by a sequence that is not eventually constant is necessarily not an ultrafilter. Every principal filter on a countable set is sequential as is every cofinite filter on a countably infinite set. The intersection of finitely many sequential filters is again sequential.
• The set [math]\displaystyle{ \mathcal{F} }[/math] of all cofinite subsets of [math]\displaystyle{ X }[/math] (meaning those sets whose complement in [math]\displaystyle{ X }[/math] is finite) is proper if and only if [math]\displaystyle{ \mathcal{F} }[/math] is infinite (or equivalently, [math]\displaystyle{ X }[/math] is infinite), in which case [math]\displaystyle{ \mathcal{F} }[/math] is a filter on [math]\displaystyle{ X }[/math] known as the Fréchet filter or the cofinite filter on [math]\displaystyle{ X. }[/math] If [math]\displaystyle{ X }[/math] is finite then [math]\displaystyle{ \mathcal{F} }[/math] is equal to the dual ideal [math]\displaystyle{ \wp(X), }[/math] which is not a filter. If [math]\displaystyle{ X }[/math] is infinite then the family [math]\displaystyle{ \{X \setminus \{x\} ~:~ x \in X\} }[/math] of complements of singleton sets is a filter subbase that generates the Fréchet filter on [math]\displaystyle{ X. }[/math] As with any family of sets over [math]\displaystyle{ X }[/math] that contains [math]\displaystyle{ \{X \setminus \{x\} ~:~ x \in X\}, }[/math] the kernel of the Fréchet filter on [math]\displaystyle{ X }[/math] is the empty set: [math]\displaystyle{ \ker \mathcal{F} = \varnothing. }[/math]
• The intersection of all elements in any non–empty family [math]\displaystyle{ \mathbb{F} \subseteq \operatorname{Filters}(X) }[/math] is itself a filter on [math]\displaystyle{ X }[/math] called the infimum or greatest lower bound of [math]\displaystyle{ \mathbb{F} \text{ in } \operatorname{Filters}(X), }[/math] which is why it may be denoted by [math]\displaystyle{ {\textstyle\bigwedge\limits_{\mathcal{F} \in \mathbb{F}}} \mathcal{F}. }[/math] Said differently, [math]\displaystyle{ \ker \mathbb{F} = {\textstyle\bigcap\limits_{\mathcal{F} \in \mathbb{F}}} \mathcal{F} \in \operatorname{Filters}(X). }[/math] Because every filter on [math]\displaystyle{ X }[/math] has [math]\displaystyle{ \{X\} }[/math] as a subset, this intersection is never empty. By definition, the infimum is the finest/largest (relative to [math]\displaystyle{ \,\subseteq\, \text{ and } \,\leq\, }[/math]) filter contained as a subset of each member of [math]\displaystyle{ \mathbb{F}. }[/math]
  • If [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{F} }[/math] are filters then their infimum in [math]\displaystyle{ \operatorname{Filters}(X) }[/math] is the filter [math]\displaystyle{ \mathcal{B} \,(\cup)\, \mathcal{F}. }[/math] If [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{F} }[/math] are prefilters then [math]\displaystyle{ \mathcal{B} \,(\cup)\, \mathcal{F} }[/math] is a prefilter that is coarser than both [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{F} }[/math] (that is, [math]\displaystyle{ \mathcal{B} \,(\cup)\, \mathcal{F} \leq \mathcal{B} \text{ and } \mathcal{B} \,(\cup)\, \mathcal{F} \leq \mathcal{F} }[/math]); indeed, it is one of the finest such prefilters, meaning that if [math]\displaystyle{ \mathcal{S} }[/math] is a prefilter such that [math]\displaystyle{ \mathcal{S} \leq \mathcal{B} \text{ and } \mathcal{S} \leq \mathcal{F} }[/math] then necessarily [math]\displaystyle{ \mathcal{S} \leq \mathcal{B} \,(\cup)\, \mathcal{F}. }[/math] More generally, if [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{F} }[/math] are non−empty families and if [math]\displaystyle{ \mathbb{S} := \{\mathcal{S} \subseteq \wp(X) ~:~ \mathcal{S} \leq \mathcal{B} \text{ and } \mathcal{S} \leq \mathcal{F}\} }[/math] then [math]\displaystyle{ \mathcal{B} \,(\cup)\, \mathcal{F} \in \mathbb{S} }[/math] and [math]\displaystyle{ \mathcal{B} \,(\cup)\, \mathcal{F} }[/math] is a greatest element of [math]\displaystyle{ (\mathbb{S}, \leq). }[/math]
• Let [math]\displaystyle{ \varnothing \neq \mathbb{F} \subseteq \operatorname{DualIdeals}(X) }[/math] and let [math]\displaystyle{ \cup \mathbb{F} = {\textstyle\bigcup\limits_{\mathcal{F} \in \mathbb{F}}} \mathcal{F}. }[/math] The supremum or least upper bound of [math]\displaystyle{ \mathbb{F} \text{ in } \operatorname{DualIdeals}(X), }[/math] denoted by [math]\displaystyle{ {\textstyle\bigvee\limits_{\mathcal{F} \in \mathbb{F}}} \mathcal{F}, }[/math] is the smallest (relative to [math]\displaystyle{ \subseteq }[/math]) dual ideal on [math]\displaystyle{ X }[/math] containing every element of [math]\displaystyle{ \mathbb{F} }[/math] as a subset; that is, it is the smallest (relative to [math]\displaystyle{ \subseteq }[/math]) dual ideal on [math]\displaystyle{ X }[/math] containing [math]\displaystyle{ \cup \mathbb{F} }[/math] as a subset. This dual ideal is [math]\displaystyle{ {\textstyle\bigvee\limits_{\mathcal{F} \in \mathbb{F}}} \mathcal{F} = \pi\left(\cup \mathbb{F}\right)^{\uparrow X}, }[/math] where [math]\displaystyle{ \pi\left(\cup \mathbb{F}\right) := \left\{F_1 \cap \cdots \cap F_n ~:~ n \in \N \text{ and every } F_i \text{ belongs to some } \mathcal{F} \in \mathbb{F}\right\} }[/math] is the π–system generated by [math]\displaystyle{ \cup \mathbb{F}. }[/math] As with any non–empty family of sets, [math]\displaystyle{ \cup \mathbb{F} }[/math] is contained in some filter on [math]\displaystyle{ X }[/math] if and only if it is a filter subbase, or equivalently, if and only if [math]\displaystyle{ {\textstyle\bigvee\limits_{\mathcal{F} \in \mathbb{F}}} \mathcal{F} = \pi\left(\cup \mathbb{F}\right)^{\uparrow X} }[/math] is a filter on [math]\displaystyle{ X, }[/math] in which case this family is the smallest (relative to [math]\displaystyle{ \subseteq }[/math]) filter on [math]\displaystyle{ X }[/math] containing every element of [math]\displaystyle{ \mathbb{F} }[/math] as a subset and necessarily [math]\displaystyle{ \mathbb{F} \subseteq \operatorname{Filters}(X). }[/math]
• Let [math]\displaystyle{ \varnothing \neq \mathbb{F} \subseteq \operatorname{Filters}(X) }[/math] and let [math]\displaystyle{ \cup \mathbb{F} = {\textstyle\bigcup\limits_{\mathcal{F} \in \mathbb{F}}} \mathcal{F}. }[/math] The supremum or least upper bound of [math]\displaystyle{ \mathbb{F} \text{ in } \operatorname{Filters}(X), }[/math] denoted by [math]\displaystyle{ {\textstyle\bigvee\limits_{\mathcal{F} \in \mathbb{F}}} \mathcal{F} }[/math] if it exists, is by definition the smallest (relative to [math]\displaystyle{ \subseteq }[/math]) filter on [math]\displaystyle{ X }[/math] containing every element of [math]\displaystyle{ \mathbb{F} }[/math] as a subset. If it exists then necessarily [math]\displaystyle{ {\textstyle\bigvee\limits_{\mathcal{F} \in \mathbb{F}}} \mathcal{F} = \pi\left(\cup \mathbb{F}\right)^{\uparrow X} }[/math] (as defined above) and [math]\displaystyle{ {\textstyle\bigvee\limits_{\mathcal{F} \in \mathbb{F}}} \mathcal{F} }[/math] will also be equal to the intersection of all filters on [math]\displaystyle{ X }[/math] containing [math]\displaystyle{ \cup \mathbb{F}. }[/math] This supremum of [math]\displaystyle{ \mathbb{F} \text{ in } \operatorname{Filters}(X) }[/math] exists if and only if the dual ideal [math]\displaystyle{ \pi\left(\cup \mathbb{F}\right)^{\uparrow X} }[/math] is a filter on [math]\displaystyle{ X. }[/math] The least upper bound of a family of filters [math]\displaystyle{ \mathbb{F} }[/math] may fail to be a filter. Indeed, if [math]\displaystyle{ X }[/math] contains at least 2 distinct elements then there exist filters [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{C} \text{ on } X }[/math] for which there does not exist a filter [math]\displaystyle{ \mathcal{F} \text{ on } X }[/math] that contains both [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{C}. }[/math] If [math]\displaystyle{ \cup \mathbb{F} }[/math] is not a filter subbase then the supremum of [math]\displaystyle{ \mathbb{F} \text{ in } \operatorname{Filters}(X) }[/math] does not exist and the same is true of its supremum in [math]\displaystyle{ \operatorname{Prefilters}(X) }[/math] but their supremum in the set of all dual ideals on [math]\displaystyle{ X }[/math] will exist (it being the degenerate filter [math]\displaystyle{ \wp(X) }[/math]).
  • If [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{F} }[/math] are prefilters (resp. filters on [math]\displaystyle{ X }[/math]) then [math]\displaystyle{ \mathcal{B} \,(\cap)\, \mathcal{F} }[/math] is a prefilter (resp. a filter) if and only if it is non–degenerate (or said differently, if and only if [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{F} }[/math] mesh), in which case it is one of the coarsest prefilters (resp. the coarsest filter) on [math]\displaystyle{ X }[/math] that is finer (with respect to [math]\displaystyle{ \,\leq }[/math]) than both [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{F}; }[/math] this means that if [math]\displaystyle{ \mathcal{S} }[/math] is any prefilter (resp. any filter) such that [math]\displaystyle{ \mathcal{B} \leq \mathcal{S} \text{ and } \mathcal{F} \leq \mathcal{S} }[/math] then necessarily [math]\displaystyle{ \mathcal{B} \,(\cap)\, \mathcal{F} \leq \mathcal{S}, }[/math] in which case it is denoted by [math]\displaystyle{ \mathcal{B} \vee \mathcal{F}. }[/math]

Other examples

• Let [math]\displaystyle{ X = \{p,1,2,3\} }[/math] and let [math]\displaystyle{ \mathcal{B} = \{\{p\}, \{p,1,2\}, \{p,1,3\}\}, }[/math] which makes [math]\displaystyle{ \mathcal{B} }[/math] a prefilter and a filter subbase that is not closed under finite intersections. Because [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter, the smallest prefilter containing [math]\displaystyle{ \mathcal{B} }[/math] is [math]\displaystyle{ \mathcal{B}. }[/math] The π–system generated by [math]\displaystyle{ \mathcal{B} }[/math] is [math]\displaystyle{ \{\{p,1\}\} \cup \mathcal{B}. }[/math] In particular, the smallest prefilter containing the filter subbase [math]\displaystyle{ \mathcal{B} }[/math] is not equal to the set of all finite intersections of sets in [math]\displaystyle{ \mathcal{B}. }[/math] The filter on [math]\displaystyle{ X }[/math] generated by [math]\displaystyle{ \mathcal{B} }[/math] is [math]\displaystyle{ \mathcal{B}^{\uparrow X} = \{S \subseteq X : p \in S\} = \{\{p\} \cup T ~:~ T \subseteq \{1,2,3\}\}. }[/math] All three of [math]\displaystyle{ \mathcal{B}, }[/math] the π–system [math]\displaystyle{ \mathcal{B} }[/math] generates, and [math]\displaystyle{ \mathcal{B}^{\uparrow X} }[/math] are examples of fixed, principal, ultra prefilters that are principal at the point [math]\displaystyle{ p; \mathcal{B}^{\uparrow X} }[/math] is also an ultrafilter on [math]\displaystyle{ X. }[/math]
• Let [math]\displaystyle{ (X, \tau) }[/math] be a topological space, [math]\displaystyle{ \mathcal{B} \subseteq \wp(X), }[/math] and define [math]\displaystyle{ \overline{\mathcal{B}} := \left\{\operatorname{cl}_X B ~:~ B \in \mathcal{B}\right\}, }[/math] where [math]\displaystyle{ \mathcal{B} }[/math] is necessarily finer than [math]\displaystyle{ \overline{\mathcal{B}}. }[/math] If [math]\displaystyle{ \mathcal{B} }[/math] is non–empty (resp. non–degenerate, a filter subbase, a prefilter, closed under finite unions) then the same is true of [math]\displaystyle{ \overline{\mathcal{B}}. }[/math] If [math]\displaystyle{ \mathcal{B} }[/math] is a filter on [math]\displaystyle{ X }[/math] then [math]\displaystyle{ \overline{\mathcal{B}} }[/math] is a prefilter but not necessarily a filter on [math]\displaystyle{ X }[/math] although [math]\displaystyle{ \left(\overline{\mathcal{B}}\right)^{\uparrow X} }[/math] is a filter on [math]\displaystyle{ X }[/math] equivalent to [math]\displaystyle{ \overline{\mathcal{B}}. }[/math]
• The set [math]\displaystyle{ \mathcal{B} }[/math] of all dense open subsets of a (non–empty) topological space [math]\displaystyle{ X }[/math] is a proper π–system and so also a prefilter. If the space is a Baire space, then the set of all countable intersections of dense open subsets is a π–system and a prefilter that is finer than [math]\displaystyle{ \mathcal{B}. }[/math] If [math]\displaystyle{ X = \R^n }[/math] (with [math]\displaystyle{ 1 \leq n \in \N }[/math]) then the set [math]\displaystyle{ \mathcal{B}_{\operatorname{LebFinite}} }[/math] of all [math]\displaystyle{ B \in \mathcal{B} }[/math] such that [math]\displaystyle{ B }[/math] has finite Lebesgue measure is a proper π–system and a free prefilter that is also a proper subset of [math]\displaystyle{ \mathcal{B}. }[/math] The prefilters [math]\displaystyle{ \mathcal{B}_{\operatorname{LebFinite}} }[/math] and [math]\displaystyle{ \mathcal{B} }[/math] are equivalent and so generate the same filter on [math]\displaystyle{ X. }[/math] The prefilter [math]\displaystyle{ \mathcal{B}_{\operatorname{LebFinite}} }[/math] is properly contained in, and not equivalent to, the prefilter consisting of all dense open subsets of [math]\displaystyle{ \R. }[/math] Since [math]\displaystyle{ X }[/math] is a Baire space, every countable intersection of sets in [math]\displaystyle{ \mathcal{B}_{\operatorname{LebFinite}} }[/math] is dense in [math]\displaystyle{ X }[/math] (and also comeagre and non–meager) so the set of all countable intersections of elements of [math]\displaystyle{ \mathcal{B}_{\operatorname{LebFinite}} }[/math] is a prefilter and π–system; it is also finer than, and not equivalent to, [math]\displaystyle{ \mathcal{B}_{\operatorname{LebFinite}}. }[/math]

Ultrafilters

There are many other characterizations of "ultrafilter" and "ultra prefilter," which are listed in the article on ultrafilters. Important properties of ultrafilters are also described in that article.

A non–empty family [math]\displaystyle{ \mathcal{B} \subseteq \wp(X) }[/math] of sets is/is an:

8. Ultra if [math]\displaystyle{ \varnothing \not\in \mathcal{B} }[/math] and any of the following equivalent conditions are satisfied:
   1. For every set [math]\displaystyle{ S \subseteq X }[/math] there exists some set [math]\displaystyle{ B \in \mathcal{B} }[/math] such that [math]\displaystyle{ B \subseteq S \text{ or } B \subseteq X \setminus S }[/math] (or equivalently, such that [math]\displaystyle{ B \cap S \text{ equals } B \text{ or } \varnothing }[/math]).
   2. For every set [math]\displaystyle{ S \subseteq {\textstyle\bigcup\limits_{B \in \mathcal{B}}} B }[/math] there exists some set [math]\displaystyle{ B \in \mathcal{B} }[/math] such that [math]\displaystyle{ B \cap S \text{ equals } B \text{ or } \varnothing. }[/math]
      • This characterization of "[math]\displaystyle{ \mathcal{B} }[/math] is ultra" does not depend on the set [math]\displaystyle{ X, }[/math] so mentioning the set [math]\displaystyle{ X }[/math] is optional when using the term "ultra."
   3. For every set [math]\displaystyle{ S }[/math] (not necessarily even a subset of [math]\displaystyle{ X }[/math]) there exists some set [math]\displaystyle{ B \in \mathcal{B} }[/math] such that [math]\displaystyle{ B \cap S \text{ equals } B \text{ or } \varnothing. }[/math]
9. Ultra prefilter if it is a prefilter that is also ultra. Equivalently, it is a filter subbase that is ultra. A prefilter [math]\displaystyle{ \mathcal{B} }[/math] is ultra if and only if it satisfies any of the following equivalent conditions:
   1. [math]\displaystyle{ \mathcal{B} }[/math] is maximal in [math]\displaystyle{ \operatorname{Prefilters}(X) }[/math] with respect to [math]\displaystyle{ \,\leq,\, }[/math] which means that [math]\displaystyle{ \text{For all } \mathcal{C} \in \operatorname{Prefilters}(X), \; \mathcal{B} \leq \mathcal{C} \; \text{ implies } \; \mathcal{C} \leq \mathcal{B}. }[/math]
   2. [math]\displaystyle{ \text{For all } \mathcal{C} \in \operatorname{Filters}(X), \; \mathcal{B} \leq \mathcal{C} \; \text{ implies } \; \mathcal{C} \leq \mathcal{B}. }[/math]
      • Although this statement is identical to that given below for ultrafilters, here [math]\displaystyle{ \mathcal{B} }[/math] is merely assumed to be a prefilter; it need not be a filter.
   3. [math]\displaystyle{ \mathcal{B}^{\uparrow X} }[/math] is ultra (and thus an ultrafilter).
   4. [math]\displaystyle{ \mathcal{B} }[/math] is equivalent to some ultrafilter.
   • A filter subbase that is ultra is necessarily a prefilter. A filter subbase is ultra if and only if it is a maximal filter subbase with respect to [math]\displaystyle{ \,\leq\, }[/math] (as above).
10. Ultrafilter on [math]\displaystyle{ X }[/math] if it is a filter on [math]\displaystyle{ X }[/math] that is ultra. Equivalently, an ultrafilter on [math]\displaystyle{ X }[/math] is a filter [math]\displaystyle{ \mathcal{B} \text{ on } X }[/math] that satisfies any of the following equivalent conditions:
    1. [math]\displaystyle{ \mathcal{B} }[/math] is generated by an ultra prefilter.
    2. For any [math]\displaystyle{ S \subseteq X, S \in \mathcal{B} \text{ or } X \setminus S \in \mathcal{B}. }[/math]
    3. [math]\displaystyle{ \mathcal{B} \cup (X \setminus \mathcal{B}) = \wp(X). }[/math] This condition can be restated as: [math]\displaystyle{ \wp(X) }[/math] is partitioned by [math]\displaystyle{ \mathcal{B} }[/math] and its dual [math]\displaystyle{ X \setminus \mathcal{B}. }[/math]
    4. For any [math]\displaystyle{ R, S \subseteq X, }[/math] if [math]\displaystyle{ R \cup S \in \mathcal{B} }[/math] then [math]\displaystyle{ R \in \mathcal{B} \text{ or } S \in \mathcal{B} }[/math] (a filter with this property is called a prime filter).
       • This property extends to any finite union of two or more sets.
    5. [math]\displaystyle{ \mathcal{B} }[/math] is a maximal filter on [math]\displaystyle{ X }[/math]; meaning that if [math]\displaystyle{ \mathcal{C} }[/math] is a filter on [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ \mathcal{B} \subseteq \mathcal{C} }[/math] then necessarily [math]\displaystyle{ \mathcal{C} = \mathcal{B} }[/math] (this equality may be replaced by [math]\displaystyle{ \mathcal{C} \subseteq \mathcal{B} \text{ or by } \mathcal{C} \leq \mathcal{B} }[/math]).
       • If [math]\displaystyle{ \mathcal{C} }[/math] is upward closed then [math]\displaystyle{ \mathcal{B} \leq \mathcal{C} \text{ if and only if } \mathcal{B} \subseteq \mathcal{C}. }[/math] So this characterization of ultrafilters as maximal filters can be restated as: [math]\displaystyle{ \text{For all } \mathcal{C} \in \operatorname{Filters}(X), \; \mathcal{B} \leq \mathcal{C} \; \text{ implies } \; \mathcal{C} \leq \mathcal{B}. }[/math]
       • Because subordination [math]\displaystyle{ \,\geq\, }[/math] is for filters the analog of "is a subnet/subsequence of" (specifically, "subnet" should mean "AA–subnet," which is defined below), this characterization of an ultrafilter as being a "maximally subordinate filter" suggests that an ultrafilter can be interpreted as being analogous to some sort of "maximally deep net" (which could, for instance, mean that "when viewed only from [math]\displaystyle{ X }[/math]" in some sense, it is indistinguishable from its subnets, as is the case with any net valued in a singleton set for example), which is an idea that is actually made rigorous by ultranets. The ultrafilter lemma is then the statement that every filter ("net") has some subordinate filter ("subnet") that is "maximally subordinate" ("maximally deep").

The ultrafilter lemma

The following important theorem is due to Alfred Tarski (1930).

The ultrafilter lemma/principle/theorem (Tarski) — Every filter on a set [math]\displaystyle{ X }[/math] is a subset of some ultrafilter on [math]\displaystyle{ X. }[/math]

A consequence of the ultrafilter lemma is that every filter is equal to the intersection of all ultrafilters containing it. Assuming the axioms of Zermelo–Fraenkel (ZF), the ultrafilter lemma follows from the Axiom of choice (in particular from Zorn's lemma) but is strictly weaker than it. The ultrafilter lemma implies the Axiom of choice for finite sets. If only dealing with Hausdorff spaces, then most basic results (as encountered in introductory courses) in Topology (such as Tychonoff's theorem for compact Hausdorff spaces and the Alexander subbase theorem) and in functional analysis (such as the Hahn–Banach theorem) can be proven using only the ultrafilter lemma; the full strength of the axiom of choice might not be needed.

Kernels

The kernel is useful in classifying properties of prefilters and other families of sets.

The kernel of a family of sets [math]\displaystyle{ \mathcal{B} }[/math] is the intersection of all sets that are elements of [math]\displaystyle{ \mathcal{B}: }[/math]

[math]\displaystyle{ \ker \mathcal{B} = \bigcap_{B \in \mathcal{B}} B }[/math]

If [math]\displaystyle{ \mathcal{B} \subseteq \wp(X) }[/math] then [math]\displaystyle{ \ker \left(\mathcal{B}^{\uparrow X}\right) = \ker \mathcal{B} }[/math] and this set is also equal to the kernel of the π–system that is generated by [math]\displaystyle{ \mathcal{B}. }[/math] In particular, if [math]\displaystyle{ \mathcal{B} }[/math] is a filter subbase then the kernels of all of the following sets are equal:

(1) [math]\displaystyle{ \mathcal{B}, }[/math] (2) the π–system generated by [math]\displaystyle{ \mathcal{B}, }[/math] and (3) the filter generated by [math]\displaystyle{ \mathcal{B}. }[/math]

If [math]\displaystyle{ f }[/math] is a map then [math]\displaystyle{ f(\ker \mathcal{B}) \subseteq \ker f(\mathcal{B}) \text{ and } f^{-1}(\ker \mathcal{B}) = \ker f^{-1}(\mathcal{B}). }[/math] Equivalent families have equal kernels. Two principal families are equivalent if and only if their kernels are equal.

Classifying families by their kernels

A family [math]\displaystyle{ \mathcal{B} }[/math] of sets is:

2. Free if [math]\displaystyle{ \ker \mathcal{B} = \varnothing, }[/math] or equivalently, if [math]\displaystyle{ \{X \setminus \{x\} ~:~ x \in X\} \subseteq \mathcal{B}^{\uparrow X}; }[/math] this can be restated as [math]\displaystyle{ \{X \setminus \{x\} ~:~ x \in X\} \leq \mathcal{B}. }[/math]
   • A filter [math]\displaystyle{ \mathcal{F} \text{ on } X }[/math] is free if and only if [math]\displaystyle{ X }[/math] is infinite and [math]\displaystyle{ \mathcal{F} }[/math] contains the Fréchet filter on [math]\displaystyle{ X }[/math] as a subset.
3. Fixed if [math]\displaystyle{ \ker \mathcal{B} \neq \varnothing }[/math] in which case, [math]\displaystyle{ \mathcal{B} }[/math] is said to be fixed by any point [math]\displaystyle{ x \in \ker \mathcal{B}. }[/math]
   • Any fixed family is necessarily a filter subbase.
4. Principal if [math]\displaystyle{ \ker \mathcal{B} \in \mathcal{B}. }[/math]
   • A proper principal family of sets is necessarily a prefilter.
5. Discrete or Principal at [math]\displaystyle{ x \in X }[/math] if [math]\displaystyle{ \{x\} = \ker \mathcal{B} \in \mathcal{B}. }[/math]
   • The principal filter at [math]\displaystyle{ x \text{ on } X }[/math] is the filter [math]\displaystyle{ \{x\}^{\uparrow X}. }[/math] A filter [math]\displaystyle{ \mathcal{F} }[/math] is principal at [math]\displaystyle{ x }[/math] if and only if [math]\displaystyle{ \mathcal{F} = \{x\}^{\uparrow X}. }[/math]
6. Countably deep if whenever [math]\displaystyle{ \mathcal{C} \subseteq \mathcal{B} }[/math] is a countable subset then [math]\displaystyle{ \ker \mathcal{C} \in \mathcal{B}. }[/math]

If [math]\displaystyle{ \mathcal{B} }[/math] is a principal filter on [math]\displaystyle{ X }[/math] then [math]\displaystyle{ \varnothing \neq \ker \mathcal{B} \in \mathcal{B} }[/math] and [math]\displaystyle{ \mathcal{B} = \{\ker \mathcal{B}\}^{\uparrow X} }[/math] and [math]\displaystyle{ \{\ker \mathcal{B}\} }[/math] is also the smallest prefilter that generates [math]\displaystyle{ \mathcal{B}. }[/math]

Family of examples: For any non–empty [math]\displaystyle{ C \subseteq \R, }[/math] the family [math]\displaystyle{ \mathcal{B}_C = \{\R \setminus (r + C) ~:~ r \in \R\} }[/math] is free but it is a filter subbase if and only if no finite union of the form [math]\displaystyle{ \left(r_1 + C\right) \cup \cdots \cup \left(r_n + C\right) }[/math] covers [math]\displaystyle{ \R, }[/math] in which case the filter that it generates will also be free. In particular, [math]\displaystyle{ \mathcal{B}_C }[/math] is a filter subbase if [math]\displaystyle{ C }[/math] is countable (for example, [math]\displaystyle{ C = \Q, \Z, }[/math] the primes), a meager set in [math]\displaystyle{ \R, }[/math] a set of finite measure, or a bounded subset of [math]\displaystyle{ \R. }[/math] If [math]\displaystyle{ C }[/math] is a singleton set then [math]\displaystyle{ \mathcal{B}_C }[/math] is a subbase for the Fréchet filter on [math]\displaystyle{ \R. }[/math]

Characterizing fixed ultra prefilters

If a family of sets [math]\displaystyle{ \mathcal{B} }[/math] is fixed (that is, [math]\displaystyle{ \ker \mathcal{B} \neq \varnothing }[/math]) then [math]\displaystyle{ \mathcal{B} }[/math] is ultra if and only if some element of [math]\displaystyle{ \mathcal{B} }[/math] is a singleton set, in which case [math]\displaystyle{ \mathcal{B} }[/math] will necessarily be a prefilter. Every principal prefilter is fixed, so a principal prefilter [math]\displaystyle{ \mathcal{B} }[/math] is ultra if and only if [math]\displaystyle{ \ker \mathcal{B} }[/math] is a singleton set.

Every filter on [math]\displaystyle{ X }[/math] that is principal at a single point is an ultrafilter, and if in addition [math]\displaystyle{ X }[/math] is finite, then there are no ultrafilters on [math]\displaystyle{ X }[/math] other than these.

The next theorem shows that every ultrafilter falls into one of two categories: either it is free or else it is a principal filter generated by a single point.

Proposition — If [math]\displaystyle{ \mathcal{F} }[/math] is an ultrafilter on [math]\displaystyle{ X }[/math] then the following are equivalent:

1. [math]\displaystyle{ \mathcal{F} }[/math] is fixed, or equivalently, not free, meaning [math]\displaystyle{ \ker \mathcal{F} \neq \varnothing. }[/math]
2. [math]\displaystyle{ \mathcal{F} }[/math] is principal, meaning [math]\displaystyle{ \ker \mathcal{F} \in \mathcal{F}. }[/math]
3. Some element of [math]\displaystyle{ \mathcal{F} }[/math] is a finite set.
4. Some element of [math]\displaystyle{ \mathcal{F} }[/math] is a singleton set.
5. [math]\displaystyle{ \mathcal{F} }[/math] is principal at some point of [math]\displaystyle{ X, }[/math] which means [math]\displaystyle{ \ker \mathcal{F} = \{x\} \in \mathcal{F} }[/math] for some [math]\displaystyle{ x \in X. }[/math]
6. [math]\displaystyle{ \mathcal{F} }[/math] does not contain the Fréchet filter on [math]\displaystyle{ X. }[/math]
7. [math]\displaystyle{ \mathcal{F} }[/math] is sequential.

Finer/coarser, subordination, and meshing

The preorder [math]\displaystyle{ \,\leq\, }[/math] that is defined below is of fundamental importance for the use of prefilters (and filters) in topology. For instance, this preorder is used to define the prefilter equivalent of "subsequence", where "[math]\displaystyle{ \mathcal{F} \geq \mathcal{C} }[/math]" can be interpreted as "[math]\displaystyle{ \mathcal{F} }[/math] is a subsequence of [math]\displaystyle{ \mathcal{C} }[/math]" (so "subordinate to" is the prefilter equivalent of "subsequence of"). It is also used to define prefilter convergence in a topological space. The definition of [math]\displaystyle{ \mathcal{B} }[/math] meshes with [math]\displaystyle{ \mathcal{C}, }[/math] which is closely related to the preorder [math]\displaystyle{ \,\leq, }[/math] is used in topology to define cluster points.

Two families of sets [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{C} }[/math] mesh and are compatible, indicated by writing [math]\displaystyle{ \mathcal{B} \# \mathcal{C}, }[/math] if [math]\displaystyle{ B \cap C \neq \varnothing \text{ for all } B \in \mathcal{B} \text{ and } C \in \mathcal{C}. }[/math] If [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{C} }[/math] do not mesh then they are dissociated. If [math]\displaystyle{ S \subseteq X \text{ and } \mathcal{B} \subseteq \wp(X) }[/math] then [math]\displaystyle{ \mathcal{B} \text{ and } S }[/math] are said to mesh if [math]\displaystyle{ \mathcal{B} \text{ and } \{S\} }[/math] mesh, or equivalently, if the trace of [math]\displaystyle{ \mathcal{B} \text{ on } S, }[/math] which is the family [math]\displaystyle{ \mathcal{B}\big\vert_S = \{B \cap S ~:~ B \in \mathcal{B}\}, }[/math] does not contain the empty set, where the trace is also called the restriction of [math]\displaystyle{ \mathcal{B} \text{ to } S. }[/math]

Declare that [math]\displaystyle{ \mathcal{C} \leq \mathcal{F}, \mathcal{F} \geq \mathcal{C}, \text{ and } \mathcal{F} \vdash \mathcal{C}, }[/math] stated as [math]\displaystyle{ \mathcal{C} }[/math] is coarser than [math]\displaystyle{ \mathcal{F} }[/math] and [math]\displaystyle{ \mathcal{F} }[/math] is finer than (or subordinate to) [math]\displaystyle{ \mathcal{C}, }[/math] if any of the following equivalent conditions hold:

1. Definition: Every [math]\displaystyle{ C \in \mathcal{C} }[/math] contains some [math]\displaystyle{ F \in \mathcal{F}. }[/math] Explicitly, this means that for every [math]\displaystyle{ C \in \mathcal{C}, }[/math] there is some [math]\displaystyle{ F \in \mathcal{F} }[/math] such that [math]\displaystyle{ F \subseteq C }[/math] (thus [math]\displaystyle{ \mathcal{C} \ni C \supseteq F \in \mathcal{F} }[/math] holds).
   • Said more briefly in plain English, [math]\displaystyle{ \mathcal{C} \leq \mathcal{F} }[/math] if every set in [math]\displaystyle{ \mathcal{C} }[/math] is larger than some set in [math]\displaystyle{ \mathcal{F}. }[/math] Here, a "larger set" means a superset.
2. [math]\displaystyle{ \{C\} \leq \mathcal{F} \text{ for every } C \in \mathcal{C}. }[/math]
   • In words, [math]\displaystyle{ \{C\} \leq \mathcal{F} }[/math] states exactly that [math]\displaystyle{ C }[/math] is larger than some set in [math]\displaystyle{ \mathcal{F}. }[/math] The equivalence of (a) and (b) follows immediately.
3. [math]\displaystyle{ \mathcal{C} \leq \mathcal{F}^{\uparrow X}, }[/math] which is equivalent to [math]\displaystyle{ \mathcal{C} \subseteq \mathcal{F}^{\uparrow X} }[/math];
4. [math]\displaystyle{ \mathcal{C}^{\uparrow X} \leq \mathcal{F} }[/math];
5. [math]\displaystyle{ \mathcal{C}^{\uparrow X} \leq \mathcal{F}^{\uparrow X}, }[/math] which is equivalent to [math]\displaystyle{ \mathcal{C}^{\uparrow X} \subseteq \mathcal{F}^{\uparrow X} }[/math];

and if in addition [math]\displaystyle{ \mathcal{F} }[/math] is upward closed, which means that [math]\displaystyle{ \mathcal{F} = \mathcal{F}^{\uparrow X}, }[/math] then this list can be extended to include:

6. [math]\displaystyle{ \mathcal{C} \subseteq \mathcal{F}. }[/math]
   • So in this case, this definition of "[math]\displaystyle{ \mathcal{F} }[/math] is finer than [math]\displaystyle{ \mathcal{C} }[/math]" would be identical to the topological definition of "finer" had [math]\displaystyle{ \mathcal{C} \text{ and } \mathcal{F} }[/math] been topologies on [math]\displaystyle{ X. }[/math]

If an upward closed family [math]\displaystyle{ \mathcal{F} }[/math] is finer than [math]\displaystyle{ \mathcal{C} }[/math] (that is, [math]\displaystyle{ \mathcal{C} \leq \mathcal{F} }[/math]) but [math]\displaystyle{ \mathcal{C} \neq \mathcal{F} }[/math] then [math]\displaystyle{ \mathcal{F} }[/math] is said to be strictly finer than [math]\displaystyle{ \mathcal{C} }[/math] and [math]\displaystyle{ \mathcal{C} }[/math] is strictly coarser than [math]\displaystyle{ \mathcal{F}. }[/math] Two families are comparable if one of them is finer than the other.

Example: If [math]\displaystyle{ x_{i_\bull} = \left(x_{i_n}\right)_{n=1}^\infty }[/math] is a subsequence of [math]\displaystyle{ x_\bull = \left(x_i\right)_{i=1}^\infty }[/math] then [math]\displaystyle{ \operatorname{Tails}\left(x_{i_\bull}\right) }[/math] is subordinate to [math]\displaystyle{ \operatorname{Tails}\left(x_\bull\right); }[/math] in symbols: [math]\displaystyle{ \operatorname{Tails}\left(x_{i_\bull}\right) \vdash \operatorname{Tails}\left(x_\bull\right) }[/math] and also [math]\displaystyle{ \operatorname{Tails}\left(x_\bull\right) \leq \operatorname{Tails}\left(x_{i_\bull}\right). }[/math] Stated in plain English, the prefilter of tails of a subsequence is always subordinate to that of the original sequence. To see this, let [math]\displaystyle{ C := x_{\geq i} \in \operatorname{Tails}\left(x_\bull\right) }[/math] be arbitrary (or equivalently, let [math]\displaystyle{ i \in \N }[/math] be arbitrary) and it remains to show that this set contains some [math]\displaystyle{ F := x_{i_{\geq n}} \in \operatorname{Tails}\left(x_{i_\bull}\right). }[/math] For the set [math]\displaystyle{ x_{\geq i} = \left\{x_i, x_{i+1}, \ldots\right\} }[/math] to contain [math]\displaystyle{ x_{i_{\geq n}} = \left\{x_{i_n}, x_{i_{n+1}}, \ldots\right\}, }[/math] it is sufficient to have [math]\displaystyle{ i \leq i_n. }[/math] Since [math]\displaystyle{ i_1 \lt i_2 \lt \cdots }[/math] are strictly increasing integers, there exists [math]\displaystyle{ n \in \N }[/math] such that [math]\displaystyle{ i_n \geq i, }[/math] and so [math]\displaystyle{ x_{\geq i} \supseteq x_{i_{\geq n}} }[/math] holds, as desired. Consequently, [math]\displaystyle{ \operatorname{TailsFilter}\left(x_\bull\right) \subseteq \operatorname{TailsFilter}\left(x_{i_\bull}\right). }[/math] The left hand side will be a strict/proper subset of the right hand side if (for instance) every point of [math]\displaystyle{ x_\bull }[/math] is unique (that is, when [math]\displaystyle{ x_\bull : \N \to X }[/math] is injective) and [math]\displaystyle{ x_{i_\bull} }[/math] is the even-indexed subsequence [math]\displaystyle{ \left(x_2, x_4, x_6, \ldots\right) }[/math] because under these conditions, every tail [math]\displaystyle{ x_{i_{\geq n}} = \left\{x_{2n}, x_{2n + 2}, x_{2n + 4}, \ldots\right\} }[/math] (for every [math]\displaystyle{ n \in \N }[/math]) of the subsequence will belong to the right hand side filter but not to the left hand side filter.

For another example, if [math]\displaystyle{ \mathcal{B} }[/math] is any family then [math]\displaystyle{ \varnothing \leq \mathcal{B} \leq \mathcal{B} \leq \{\varnothing\} }[/math] always holds and furthermore, [math]\displaystyle{ \{\varnothing\} \leq \mathcal{B} \text{ if and only if } \varnothing \in \mathcal{B}. }[/math]

A non-empty family that is coarser than a filter subbase must itself be a filter subbase. Every filter subbase is coarser than both the π–system that it generates and the filter that it generates.

If [math]\displaystyle{ \mathcal{C} \text{ and } \mathcal{F} }[/math] are families such that [math]\displaystyle{ \mathcal{C} \leq \mathcal{F}, }[/math] the family [math]\displaystyle{ \mathcal{C} }[/math] is ultra, and [math]\displaystyle{ \varnothing \not\in \mathcal{F}, }[/math] then [math]\displaystyle{ \mathcal{F} }[/math] is necessarily ultra. It follows that any family that is equivalent to an ultra family will necessarily be ultra. In particular, if [math]\displaystyle{ \mathcal{C} }[/math] is a prefilter then either both [math]\displaystyle{ \mathcal{C} }[/math] and the filter [math]\displaystyle{ \mathcal{C}^{\uparrow X} }[/math] it generates are ultra or neither one is ultra.

The relation [math]\displaystyle{ \,\leq\, }[/math] is reflexive and transitive, which makes it into a preorder on [math]\displaystyle{ \wp(\wp(X)). }[/math] The relation [math]\displaystyle{ \,\leq\, \text{ on } \operatorname{Filters}(X) }[/math] is antisymmetric but if [math]\displaystyle{ X }[/math] has more than one point then it is not symmetric.

Equivalent families of sets

The preorder [math]\displaystyle{ \,\leq\, }[/math] induces its canonical equivalence relation on [math]\displaystyle{ \wp(\wp(X)), }[/math] where for all [math]\displaystyle{ \mathcal{B}, \mathcal{C} \in \wp(\wp(X)), }[/math] [math]\displaystyle{ \mathcal{B} }[/math] is equivalent to [math]\displaystyle{ \mathcal{C} }[/math] if any of the following equivalent conditions hold:

1. [math]\displaystyle{ \mathcal{C} \leq \mathcal{B} \text{ and } \mathcal{B} \leq \mathcal{C}. }[/math]
2. The upward closures of [math]\displaystyle{ \mathcal{C} \text{ and } \mathcal{B} }[/math] are equal.

Two upward closed (in [math]\displaystyle{ X }[/math]) subsets of [math]\displaystyle{ \wp(X) }[/math] are equivalent if and only if they are equal. If [math]\displaystyle{ \mathcal{B} \subseteq \wp(X) }[/math] then necessarily [math]\displaystyle{ \varnothing \leq \mathcal{B} \leq \wp(X) }[/math] and [math]\displaystyle{ \mathcal{B} }[/math] is equivalent to [math]\displaystyle{ \mathcal{B}^{\uparrow X}. }[/math] Every equivalence class other than [math]\displaystyle{ \{\varnothing\} }[/math] contains a unique representative (that is, element of the equivalence class) that is upward closed in [math]\displaystyle{ X. }[/math]

Properties preserved between equivalent families

Let [math]\displaystyle{ \mathcal{B}, \mathcal{C} \in \wp(\wp(X)) }[/math] be arbitrary and let [math]\displaystyle{ \mathcal{F} }[/math] be any family of sets. If [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{C} }[/math] are equivalent (which implies that [math]\displaystyle{ \ker \mathcal{B} = \ker \mathcal{C} }[/math]) then for each of the statements/properties listed below, either it is true of both [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{C} }[/math] or else it is false of both [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{C} }[/math]:

1. Not empty
2. Proper (that is, [math]\displaystyle{ \varnothing }[/math] is not an element)
   • Moreover, any two degenerate families are necessarily equivalent.
3. Filter subbase
4. Prefilter
   • In which case [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{C} }[/math] generate the same filter on [math]\displaystyle{ X }[/math] (that is, their upward closures in [math]\displaystyle{ X }[/math] are equal).
5. Free
6. Principal
7. Ultra
8. Is equal to the trivial filter [math]\displaystyle{ \{X\} }[/math]
   • In words, this means that the only subset of [math]\displaystyle{ \wp(X) }[/math] that is equivalent to the trivial filter is the trivial filter. In general, this conclusion of equality does not extend to non−trivial filters (one exception is when both families are filters).
9. Meshes with [math]\displaystyle{ \mathcal{F} }[/math]
10. Is finer than [math]\displaystyle{ \mathcal{F} }[/math]
11. Is coarser than [math]\displaystyle{ \mathcal{F} }[/math]
12. Is equivalent to [math]\displaystyle{ \mathcal{F} }[/math]

Missing from the above list is the word "filter" because this property is not preserved by equivalence. However, if [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{C} }[/math] are filters on [math]\displaystyle{ X, }[/math] then they are equivalent if and only if they are equal; this characterization does not extend to prefilters.

Equivalence of prefilters and filter subbases

If [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter on [math]\displaystyle{ X }[/math] then the following families are always equivalent to each other:

1. [math]\displaystyle{ \mathcal{B} }[/math];
2. the π–system generated by [math]\displaystyle{ \mathcal{B} }[/math];
3. the filter on [math]\displaystyle{ X }[/math] generated by [math]\displaystyle{ \mathcal{B} }[/math];

and moreover, these three families all generate the same filter on [math]\displaystyle{ X }[/math] (that is, the upward closures in [math]\displaystyle{ X }[/math] of these families are equal).

In particular, every prefilter is equivalent to the filter that it generates. By transitivity, two prefilters are equivalent if and only if they generate the same filter. Every prefilter is equivalent to exactly one filter on [math]\displaystyle{ X, }[/math] which is the filter that it generates (that is, the prefilter's upward closure). Said differently, every equivalence class of prefilters contains exactly one representative that is a filter. In this way, filters can be considered as just being distinguished elements of these equivalence classes of prefilters.

A filter subbase that is not also a prefilter cannot be equivalent to the prefilter (or filter) that it generates. In contrast, every prefilter is equivalent to the filter that it generates. This is why prefilters can, by and large, be used interchangeably with the filters that they generate while filter subbases cannot.

Set theoretic properties and constructions relevant to topology

Trace and meshing

If [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter (resp. filter) on [math]\displaystyle{ X \text{ and } S \subseteq X }[/math] then the trace of [math]\displaystyle{ \mathcal{B} \text{ on } S, }[/math] which is the family [math]\displaystyle{ \mathcal{B}\big\vert_S := \mathcal{B} (\cap) \{S\}, }[/math] is a prefilter (resp. a filter) if and only if [math]\displaystyle{ \mathcal{B} \text{ and } S }[/math] mesh (that is, [math]\displaystyle{ \varnothing \not\in \mathcal{B} (\cap) \{S\} }[/math]), in which case the trace of [math]\displaystyle{ \mathcal{B} \text{ on } S }[/math] is said to be induced by [math]\displaystyle{ S }[/math]. The trace is always finer than the original family; that is, [math]\displaystyle{ \mathcal{B} \leq \mathcal{B}\big\vert_S. }[/math] If [math]\displaystyle{ \mathcal{B} }[/math] is ultra and if [math]\displaystyle{ \mathcal{B} \text{ and } S }[/math] mesh then the trace [math]\displaystyle{ \mathcal{B}\big\vert_S }[/math] is ultra. If [math]\displaystyle{ \mathcal{B} }[/math] is an ultrafilter on [math]\displaystyle{ X }[/math] then the trace of [math]\displaystyle{ \mathcal{B} \text{ on } S }[/math] is a filter on [math]\displaystyle{ S }[/math] if and only if [math]\displaystyle{ S \in \mathcal{B}. }[/math]

For example, suppose that [math]\displaystyle{ \mathcal{B} }[/math] is a filter on [math]\displaystyle{ X \text{ and } S \subseteq X }[/math] is such that [math]\displaystyle{ S \neq X \text{ and } X \setminus S \not\in \mathcal{B}. }[/math] Then [math]\displaystyle{ \mathcal{B} \text{ and } S }[/math] mesh and [math]\displaystyle{ \mathcal{B} \cup \{S\} }[/math] generates a filter on [math]\displaystyle{ X }[/math] that is strictly finer than [math]\displaystyle{ \mathcal{B}. }[/math]

When prefilters mesh

Given non–empty families [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{C}, }[/math] the family [math]\displaystyle{ \mathcal{B} (\cap) \mathcal{C} := \{B \cap C ~:~ B \in \mathcal{B} \text{ and } C \in \mathcal{C}\} }[/math] satisfies [math]\displaystyle{ \mathcal{C} \leq \mathcal{B} (\cap) \mathcal{C} }[/math] and [math]\displaystyle{ \mathcal{B} \leq \mathcal{B} (\cap) \mathcal{C}. }[/math] If [math]\displaystyle{ \mathcal{B} (\cap) \mathcal{C} }[/math] is proper (resp. a prefilter, a filter subbase) then this is also true of both [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{C}. }[/math] In order to make any meaningful deductions about [math]\displaystyle{ \mathcal{B} (\cap) \mathcal{C} }[/math] from [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{C}, \mathcal{B} (\cap) \mathcal{C} }[/math] needs to be proper (that is, [math]\displaystyle{ \varnothing \not\in \mathcal{B} (\cap) \mathcal{C}, }[/math] which is the motivation for the definition of "mesh". In this case, [math]\displaystyle{ \mathcal{B} (\cap) \mathcal{C} }[/math] is a prefilter (resp. filter subbase) if and only if this is true of both [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{C}. }[/math] Said differently, if [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{C} }[/math] are prefilters then they mesh if and only if [math]\displaystyle{ \mathcal{B} (\cap) \mathcal{C} }[/math] is a prefilter. Generalizing gives a well known characterization of "mesh" entirely in terms of subordination (that is, [math]\displaystyle{ \,\leq\, }[/math]):

Two prefilters (resp. filter subbases) [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{C} }[/math] mesh if and only if there exists a prefilter (resp. filter subbase) [math]\displaystyle{ \mathcal{F} }[/math] such that [math]\displaystyle{ \mathcal{C} \leq \mathcal{F} }[/math] and [math]\displaystyle{ \mathcal{B} \leq \mathcal{F}. }[/math]

If the least upper bound of two filters [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{C} }[/math] exists in [math]\displaystyle{ \operatorname{Filters}(X) }[/math] then this least upper bound is equal to [math]\displaystyle{ \mathcal{B} (\cap) \mathcal{C}. }[/math]

Images and preimages under functions

Throughout, [math]\displaystyle{ f : X \to Y \text{ and } g : Y \to Z }[/math] will be maps between non–empty sets.

Images of prefilters

Let [math]\displaystyle{ \mathcal{B} \subseteq \wp(Y). }[/math] Many of the properties that [math]\displaystyle{ \mathcal{B} }[/math] may have are preserved under images of maps; notable exceptions include being upward closed, being closed under finite intersections, and being a filter, which are not necessarily preserved.

Explicitly, if one of the following properties is true of [math]\displaystyle{ \mathcal{B} \text{ on } Y, }[/math] then it will necessarily also be true of [math]\displaystyle{ g(\mathcal{B}) \text{ on } g(Y) }[/math] (although possibly not on the codomain [math]\displaystyle{ Z }[/math] unless [math]\displaystyle{ g }[/math] is surjective): ultra, ultrafilter, filter, prefilter, filter subbase, dual ideal, upward closed, proper/non–degenerate, ideal, closed under finite unions, downward closed, directed upward. Moreover, if [math]\displaystyle{ \mathcal{B} \subseteq \wp(Y) }[/math] is a prefilter then so are both [math]\displaystyle{ g(\mathcal{B}) \text{ and } g^{-1}(g(\mathcal{B})). }[/math] The image under a map [math]\displaystyle{ f : X \to Y }[/math] of an ultra set [math]\displaystyle{ \mathcal{B} \subseteq \wp(X) }[/math] is again ultra and if [math]\displaystyle{ \mathcal{B} }[/math] is an ultra prefilter then so is [math]\displaystyle{ f(\mathcal{B}). }[/math]

If [math]\displaystyle{ \mathcal{B} }[/math] is a filter then [math]\displaystyle{ g(\mathcal{B}) }[/math] is a filter on the range [math]\displaystyle{ g(Y), }[/math] but it is a filter on the codomain [math]\displaystyle{ Z }[/math] if and only if [math]\displaystyle{ g }[/math] is surjective. Otherwise it is just a prefilter on [math]\displaystyle{ Z }[/math] and its upward closure must be taken in [math]\displaystyle{ Z }[/math] to obtain a filter. The upward closure of [math]\displaystyle{ g(\mathcal{B}) \text{ in } Z }[/math] is [math]\displaystyle{ g(\mathcal{B})^{\uparrow Z} = \left\{S \subseteq Z ~:~ B \subseteq g^{-1}(S) \text{ for some } B \in \mathcal{B}\right\} }[/math] where if [math]\displaystyle{ \mathcal{B} }[/math] is upward closed in [math]\displaystyle{ Y }[/math] (that is, a filter) then this simplifies to: [math]\displaystyle{ g(\mathcal{B})^{\uparrow Z} = \left\{S \subseteq Z ~:~ g^{-1}(S) \in \mathcal{B}\right\}. }[/math]

If [math]\displaystyle{ X \subseteq Y }[/math] then taking [math]\displaystyle{ g }[/math] to be the inclusion map [math]\displaystyle{ X \to Y }[/math] shows that any prefilter (resp. ultra prefilter, filter subbase) on [math]\displaystyle{ X }[/math] is also a prefilter (resp. ultra prefilter, filter subbase) on [math]\displaystyle{ Y. }[/math]

Preimages of prefilters

Let [math]\displaystyle{ \mathcal{B} \subseteq \wp(Y). }[/math] Under the assumption that [math]\displaystyle{ f : X \to Y }[/math] is surjective:

[math]\displaystyle{ f^{-1}(\mathcal{B}) }[/math] is a prefilter (resp. filter subbase, π–system, closed under finite unions, proper) if and only if this is true of [math]\displaystyle{ \mathcal{B}. }[/math]

However, if [math]\displaystyle{ \mathcal{B} }[/math] is an ultrafilter on [math]\displaystyle{ Y }[/math] then even if [math]\displaystyle{ f }[/math] is surjective (which would make [math]\displaystyle{ f^{-1}(\mathcal{B}) }[/math] a prefilter), it is nevertheless still possible for the prefilter [math]\displaystyle{ f^{-1}(\mathcal{B}) }[/math] to be neither ultra nor a filter on [math]\displaystyle{ X. }[/math] 

If [math]\displaystyle{ f : X \to Y }[/math] is not surjective then denote the trace of [math]\displaystyle{ \mathcal{B} \text{ on } f(X) }[/math] by [math]\displaystyle{ \mathcal{B}\big\vert_{f(X)}, }[/math] where in this case particular case the trace satisfies: [math]\displaystyle{ \mathcal{B}\big\vert_{f(X)} = f\left(f^{-1}(\mathcal{B})\right) }[/math] and consequently also: [math]\displaystyle{ f^{-1}(\mathcal{B}) = f^{-1}\left(\mathcal{B}\big\vert_{f(X)}\right). }[/math]

This last equality and the fact that the trace [math]\displaystyle{ \mathcal{B}\big\vert_{f(X)} }[/math] is a family of sets over [math]\displaystyle{ f(X) }[/math] means that to draw conclusions about [math]\displaystyle{ f^{-1}(\mathcal{B}), }[/math] the trace [math]\displaystyle{ \mathcal{B}\big\vert_{f(X)} }[/math] can be used in place of [math]\displaystyle{ \mathcal{B} }[/math] and the surjection [math]\displaystyle{ f : X \to f(X) }[/math] can be used in place of [math]\displaystyle{ f : X \to Y. }[/math] For example:

[math]\displaystyle{ f^{-1}(\mathcal{B}) }[/math] is a prefilter (resp. filter subbase, π–system, proper) if and only if this is true of [math]\displaystyle{ \mathcal{B}\big\vert_{f(X)}. }[/math]

In this way, the case where [math]\displaystyle{ f }[/math] is not (necessarily) surjective can be reduced down to the case of a surjective function (which is a case that was described at the start of this subsection).

Even if [math]\displaystyle{ \mathcal{B} }[/math] is an ultrafilter on [math]\displaystyle{ Y, }[/math] if [math]\displaystyle{ f }[/math] is not surjective then it is nevertheless possible that [math]\displaystyle{ \varnothing \in \mathcal{B}\big\vert_{f(X)}, }[/math] which would make [math]\displaystyle{ f^{-1}(\mathcal{B}) }[/math] degenerate as well. The next characterization shows that degeneracy is the only obstacle. If [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter then the following are equivalent:

1. [math]\displaystyle{ f^{-1}(\mathcal{B}) }[/math] is a prefilter;
2. [math]\displaystyle{ \mathcal{B}\big\vert_{f(X)} }[/math] is a prefilter;
3. [math]\displaystyle{ \varnothing \not\in \mathcal{B}\big\vert_{f(X)} }[/math];
4. [math]\displaystyle{ \mathcal{B} }[/math] meshes with [math]\displaystyle{ f(X) }[/math]

and moreover, if [math]\displaystyle{ f^{-1}(\mathcal{B}) }[/math] is a prefilter then so is [math]\displaystyle{ f\left(f^{-1}(\mathcal{B})\right). }[/math]

If [math]\displaystyle{ S \subseteq Y }[/math] and if [math]\displaystyle{ \operatorname{In} : S \to Y }[/math] denotes the inclusion map then the trace of [math]\displaystyle{ \mathcal{B} \text{ on } S }[/math] is equal to [math]\displaystyle{ \operatorname{In}^{-1}(\mathcal{B}). }[/math] This observation allows the results in this subsection to be applied to investigating the trace on a set.

Subordination is preserved by images and preimages

The relation [math]\displaystyle{ \,\leq\, }[/math] is preserved under both images and preimages of families of sets. This means that for any families [math]\displaystyle{ \mathcal{C} \text{ and } \mathcal{F}, }[/math] [math]\displaystyle{ \mathcal{C} \leq \mathcal{F} \quad \text{ implies } \quad g(\mathcal{C}) \leq g(\mathcal{F}) \quad \text{ and } \quad f^{-1}(\mathcal{C}) \leq f^{-1}(\mathcal{F}). }[/math]

Moreover, the following relations always hold for any family of sets [math]\displaystyle{ \mathcal{C} }[/math]: [math]\displaystyle{ \mathcal{C} \leq f\left(f^{-1}(\mathcal{C})\right) }[/math] where equality will hold if [math]\displaystyle{ f }[/math] is surjective. Furthermore, [math]\displaystyle{ f^{-1}(\mathcal{C}) = f^{-1}\left(f\left(f^{-1}(\mathcal{C})\right)\right) \quad \text{ and } \quad g(\mathcal{C}) = g\left(g^{-1}(g(\mathcal{C}))\right). }[/math]

If [math]\displaystyle{ \mathcal{B} \subseteq \wp(X) \text{ and } \mathcal{C} \subseteq \wp(Y) }[/math] then [math]\displaystyle{ f(\mathcal{B}) \leq \mathcal{C} \quad \text{ if and only if } \quad \mathcal{B} \leq f^{-1}(\mathcal{C}) }[/math] and [math]\displaystyle{ g^{-1}(g(\mathcal{C})) \leq \mathcal{C} }[/math] where equality will hold if [math]\displaystyle{ g }[/math] is injective.

Products of prefilters

Suppose [math]\displaystyle{ X_\bull = \left(X_i\right)_{i \in I} }[/math] is a family of one or more non–empty sets, whose product will be denoted by [math]\displaystyle{ {\textstyle\prod_{}} X_\bull := {\textstyle\prod\limits_{i \in I}} X_i, }[/math] and for every index [math]\displaystyle{ i \in I, }[/math] let [math]\displaystyle{ \Pr{}_{X_i} : \prod X_\bull \to X_i }[/math] denote the canonical projection. Let [math]\displaystyle{ \mathcal{B}_\bull := \left(\mathcal{B}_i\right)_{i \in I} }[/math] be non−empty families, also indexed by [math]\displaystyle{ I, }[/math] such that [math]\displaystyle{ \mathcal{B}_i \subseteq \wp\left(X_i\right) }[/math] for each [math]\displaystyle{ i \in I. }[/math] The product of the families [math]\displaystyle{ \mathcal{B}_\bull }[/math] is defined identically to how the basic open subsets of the product topology are defined (had all of these [math]\displaystyle{ \mathcal{B}_i }[/math] been topologies). That is, both the notations [math]\displaystyle{ \prod_{} \mathcal{B}_\bull = \prod_{i \in I} \mathcal{B}_i }[/math] denote the family of all cylinder subsets [math]\displaystyle{ {\textstyle\prod\limits_{i \in I}} S_i \subseteq {\textstyle\prod} X_\bull }[/math] such that [math]\displaystyle{ S_i = X_i }[/math] for all but finitely many [math]\displaystyle{ i \in I }[/math] and where [math]\displaystyle{ S_i \in \mathcal{B}_i }[/math] for any one of these finitely many exceptions (that is, for any [math]\displaystyle{ i }[/math] such that [math]\displaystyle{ S_i \neq X_i, }[/math] necessarily [math]\displaystyle{ S_i \in \mathcal{B}_i }[/math]). When every [math]\displaystyle{ \mathcal{B}_i }[/math] is a filter subbase then the family [math]\displaystyle{ {\textstyle\bigcup\limits_{i \in I}} \Pr{}_{X_i}^{-1} \left(\mathcal{B}_i\right) }[/math] is a filter subbase for the filter on [math]\displaystyle{ {\textstyle\prod} X_\bull }[/math] generated by [math]\displaystyle{ \mathcal{B}_\bull. }[/math] If [math]\displaystyle{ {\textstyle\prod} \mathcal{B}_\bull }[/math] is a filter subbase then the filter on [math]\displaystyle{ {\textstyle\prod} X_\bull }[/math] that it generates is called the filter generated by [math]\displaystyle{ \mathcal{B}_\bull }[/math]. If every [math]\displaystyle{ \mathcal{B}_i }[/math] is a prefilter on [math]\displaystyle{ X_i }[/math] then [math]\displaystyle{ {\textstyle\prod} \mathcal{B}_\bull }[/math] will be a prefilter on [math]\displaystyle{ {\textstyle\prod} X_\bull }[/math] and moreover, this prefilter is equal to the coarsest prefilter [math]\displaystyle{ \mathcal{F} \text{ on } {\textstyle\prod} X_\bull }[/math] such that [math]\displaystyle{ \Pr{}_{X_i} (\mathcal{F}) = \mathcal{B}_i }[/math] for every [math]\displaystyle{ i \in I. }[/math] However, [math]\displaystyle{ {\textstyle\prod} \mathcal{B}_\bull }[/math] may fail to be a filter on [math]\displaystyle{ {\textstyle\prod} X_\bull }[/math] even if every [math]\displaystyle{ \mathcal{B}_i }[/math] is a filter on [math]\displaystyle{ X_i. }[/math]

Convergence, limits, and cluster points

Throughout, [math]\displaystyle{ (X, \tau) }[/math] is a topological space.

Prefilters vs. filters

With respect to maps and subsets, the property of being a prefilter is in general more well behaved and better preserved than the property of being a filter. For instance, the image of a prefilter under some map is again a prefilter; but the image of a filter under a non–surjective map is never a filter on the codomain, although it will be a prefilter. The situation is the same with preimages under non–injective maps (even if the map is surjective). If [math]\displaystyle{ S \subseteq X }[/math] is a proper subset then any filter on [math]\displaystyle{ S }[/math] will not be a filter on [math]\displaystyle{ X, }[/math] although it will be a prefilter.

One advantage that filters have is that they are distinguished representatives of their equivalence class (relative to [math]\displaystyle{ \,\leq }[/math]), meaning that any equivalence class of prefilters contains a unique filter. This property may be useful when dealing with equivalence classes of prefilters (for instance, they are useful in the construction of completions of uniform spaces via Cauchy filters). The many properties that characterize ultrafilters are also often useful. They are used to, for example, construct the Stone–Čech compactification. The use of ultrafilters generally requires that the ultrafilter lemma be assumed. But in the many fields where the axiom of choice (or the Hahn–Banach theorem) is assumed, the ultrafilter lemma necessarily holds and does not require an addition assumption.

A note on intuition

Suppose that [math]\displaystyle{ \mathcal{F} }[/math] is a non–principal filter on an infinite set [math]\displaystyle{ X. }[/math] [math]\displaystyle{ \mathcal{F} }[/math] has one "upward" property (that of being closed upward) and one "downward" property (that of being directed downward). Starting with any [math]\displaystyle{ F_0 \in \mathcal{F}, }[/math] there always exists some [math]\displaystyle{ F_1 \in \mathcal{F} }[/math] that is a proper subset of [math]\displaystyle{ F_0 }[/math]; this may be continued ad infinitum to get a sequence [math]\displaystyle{ F_0 \supsetneq F_1 \supsetneq \cdots }[/math] of sets in [math]\displaystyle{ \mathcal{F} }[/math] with each [math]\displaystyle{ F_{i+1} }[/math] being a proper subset of [math]\displaystyle{ F_i. }[/math] The same is not true going "upward", for if [math]\displaystyle{ F_0 = X \in \mathcal{F} }[/math] then there is no set in [math]\displaystyle{ \mathcal{F} }[/math] that contains [math]\displaystyle{ X }[/math] as a proper subset. Thus when it comes to limiting behavior (which is a topic central to the field of topology), going "upward" leads to a dead end, while going "downward" is typically fruitful. So to gain understanding and intuition about how filters (and prefilter) relate to concepts in topology, the "downward" property is usually the one to concentrate on. This is also why so many topological properties can be described by using only prefilters, rather than requiring filters (which only differ from prefilters in that they are also upward closed). The "upward" property of filters is less important for topological intuition but it is sometimes useful to have for technical reasons. For example, with respect to [math]\displaystyle{ \,\subseteq, }[/math] every filter subbase is contained in a unique smallest filter but there may not exist a unique smallest prefilter containing it.

Limits and convergence

A family [math]\displaystyle{ \mathcal{B} }[/math] is said to converge in [math]\displaystyle{ (X, \tau) }[/math] to a point or subset [math]\displaystyle{ x }[/math] of [math]\displaystyle{ X }[/math] if [math]\displaystyle{ \mathcal{B} \geq \mathcal{N}(x). }[/math] Explicitly, [math]\displaystyle{ \mathcal{N}(x) \leq \mathcal{B} }[/math] means that every neighborhood [math]\displaystyle{ N \text{ of } x }[/math] contains some [math]\displaystyle{ B \in \mathcal{B} }[/math] as a subset (that is, [math]\displaystyle{ B \subseteq N }[/math]); thus the following then holds: [math]\displaystyle{ \mathcal{N} \ni N \supseteq B \in \mathcal{B}. }[/math] In words, a family converges to a point or subset [math]\displaystyle{ x }[/math] if and only if it is finer than the neighborhood filter at [math]\displaystyle{ x. }[/math] A family [math]\displaystyle{ \mathcal{B} }[/math] converging to a point or subset [math]\displaystyle{ x }[/math] may be indicated by writing [math]\displaystyle{ \mathcal{B} \to x \text{ or } \lim \mathcal{B} \to x \text{ in } X }[/math] and saying that [math]\displaystyle{ x }[/math] is a limit of [math]\displaystyle{ \mathcal{B} \text{ in } X; }[/math] if this limit [math]\displaystyle{ x }[/math] is a point (and not a subset), then [math]\displaystyle{ x }[/math] is also called a limit point. As usual, [math]\displaystyle{ \lim \mathcal{B} = x }[/math] is defined to mean that [math]\displaystyle{ \mathcal{B} \to x }[/math] and [math]\displaystyle{ x \in X }[/math] is the only limit point of [math]\displaystyle{ \mathcal{B}; }[/math] that is, if also [math]\displaystyle{ \mathcal{B} \to z \text{ then } z = x. }[/math] (If the notation "[math]\displaystyle{ \lim \mathcal{B} = x }[/math]" did not also require that the limit point [math]\displaystyle{ x }[/math] be unique then the equals sign = would no longer be guaranteed to be transitive). The set of all limit points of [math]\displaystyle{ \mathcal{B} }[/math] is denoted by [math]\displaystyle{ \lim {}_X \mathcal{B} \text{ or } \lim \mathcal{B}. }[/math]

In the above definitions, it suffices to check that [math]\displaystyle{ \mathcal{B} }[/math] is finer than some (or equivalently, finer than every) neighborhood base in [math]\displaystyle{ (X, \tau) }[/math] of the point or set (for example, such as [math]\displaystyle{ \tau(x) = \{U \in \tau : x \in U\} }[/math] or [math]\displaystyle{ \tau(S) = {\textstyle\bigcap\limits_{s \in S}} \tau(s) }[/math] when [math]\displaystyle{ S \neq \varnothing }[/math]).

Examples

If [math]\displaystyle{ X := \R^n }[/math] is Euclidean space and [math]\displaystyle{ \|x\| }[/math] denotes the Euclidean norm (which is the distance from the origin, defined as usual), then all of the following families converge to the origin:

1. the prefilter [math]\displaystyle{ \{B_r(0) : 0 \lt r \leq 1\} }[/math] of all open balls centered at the origin, where [math]\displaystyle{ B_r(z) = \{x : \|x - z\| \lt r\}. }[/math]
2. the prefilter [math]\displaystyle{ \{B_{\leq r}(0) : 0 \lt r \leq 1\} }[/math] of all closed balls centered at the origin, where [math]\displaystyle{ B_{\leq r}(z) = \{x : \|x - z\| \leq r\}. }[/math] This prefilter is equivalent to the one above.
3. the prefilter [math]\displaystyle{ \{R \cap B_{\leq r}(0) : 0 \lt r \leq 1\} }[/math] where [math]\displaystyle{ R = S_1 \cup S_{1/2} \cup S_{1/3} \cup \cdots }[/math] is a union of spheres [math]\displaystyle{ S_r = \{x : \|x\| = r\} }[/math] centered at the origin having progressively smaller radii. This family consists of the sets [math]\displaystyle{ S_{1/n} \cup S_{1/(n+1)} \cup S_{1/(n+2)} \cup \cdots }[/math] as [math]\displaystyle{ n }[/math] ranges over the positive integers.
4. any of the families above but with the radius [math]\displaystyle{ r }[/math] ranging over [math]\displaystyle{ 1, \, 1/2, \, 1/3, \, 1/4, \ldots }[/math] (or over any other positive decreasing sequence) instead of over all positive reals.
   • Drawing or imagining any one of these sequences of sets when [math]\displaystyle{ X = \R^2 }[/math] has dimension [math]\displaystyle{ n = 2 }[/math] suggests that intuitively, these sets "should" converge to the origin (and indeed they do). This is the intuition that the above definition of a "convergent prefilter" make rigorous.

Although [math]\displaystyle{ \|\cdot\| }[/math] was assumed to be the Euclidean norm, the example above remains valid for any other norm on [math]\displaystyle{ \R^n. }[/math]

The one and only limit point in [math]\displaystyle{ X := \R }[/math] of the free prefilter [math]\displaystyle{ \{(0, r) : r \gt 0\} }[/math] is [math]\displaystyle{ 0 }[/math] since every open ball around the origin contains some open interval of this form. The fixed prefilter [math]\displaystyle{ \mathcal{B} := \{[0, 1 + r) : r \gt 0\} }[/math] does not converges in [math]\displaystyle{ \R }[/math] to any point and so [math]\displaystyle{ \lim \mathcal{B} = \varnothing, }[/math] although [math]\displaystyle{ \mathcal{B} }[/math] does converge to the set [math]\displaystyle{ \ker \mathcal{B} = [0, 1] }[/math] since [math]\displaystyle{ \mathcal{N}([0, 1]) \leq \mathcal{B}. }[/math] However, not every fixed prefilter converges to its kernel. For instance, the fixed prefilter [math]\displaystyle{ \{[0, 1 + r) \cup (1 + 1/r, \infty) : r \gt 0\} }[/math] also has kernel [math]\displaystyle{ [0, 1] }[/math] but does not converges (in [math]\displaystyle{ \R }[/math]) to it.

The free prefilter [math]\displaystyle{ (\R, \infty) := \{(r, \infty) : r \in \R\} }[/math] of intervals does not converge (in [math]\displaystyle{ \R }[/math]) to any point, and it converges to a subset [math]\displaystyle{ S \subseteq \R }[/math] if and only if [math]\displaystyle{ S \in (\R, \infty)^{\uparrow \R} }[/math] (that is, if and only if the set contains some interval of the form [math]\displaystyle{ (r, \infty) }[/math] as a subset). The same is also true of the prefilter [math]\displaystyle{ [\R, \infty) := \{[r, \infty) : r \in \R\} }[/math] because it is equivalent to [math]\displaystyle{ (\R, \infty) }[/math] and equivalent families have the same limits. In fact, if [math]\displaystyle{ \mathcal{B} }[/math] is any prefilter in any topological space [math]\displaystyle{ X }[/math] then for every [math]\displaystyle{ S \in \mathcal{B}^{\uparrow X}, }[/math] [math]\displaystyle{ \mathcal{B} \to S; }[/math] in particular, every prefilter converges to the set [math]\displaystyle{ X. }[/math] More generally, because the only neighborhood of [math]\displaystyle{ X }[/math] is itself (that is, [math]\displaystyle{ \mathcal{N}(X) = \{X\} }[/math]), every non-empty family (including every filter subbase) converges to [math]\displaystyle{ X. }[/math]

For any point or subset [math]\displaystyle{ x, }[/math] its neighborhood filter [math]\displaystyle{ \mathcal{N}(x) \to x }[/math] always converges to [math]\displaystyle{ x. }[/math] More generally, any neighborhood basis at [math]\displaystyle{ x }[/math] converges to [math]\displaystyle{ x. }[/math] In any topological space, a family converges to a point [math]\displaystyle{ x }[/math] if and only if it converges to the singleton set [math]\displaystyle{ \{x\}. }[/math] When a space carries the indiscrete topology then every non-empty family converges to every non-empty subset (and thus also to every point since singleton sets are non-empty). A point [math]\displaystyle{ x }[/math] is always a limit point of the principle ultra prefilter [math]\displaystyle{ \{\{x\}\} }[/math] and of the ultrafilter that it generates. The empty family [math]\displaystyle{ \mathcal{B} = \varnothing }[/math] does not converge to any point nor to any set. Because the empty set is always an open neighborhood of itself, a family [math]\displaystyle{ \mathcal{B} }[/math] converges to [math]\displaystyle{ \varnothing }[/math] if and only if [math]\displaystyle{ \varnothing \in \mathcal{B}. }[/math] Thus no filter, prefilter, or other non-degenerate family can converge to the empty set.

If [math]\displaystyle{ S \neq \varnothing }[/math] is a non-empty subset then [math]\displaystyle{ \mathcal{N}(S) = {\textstyle\bigcap\limits_{s \in S}} \mathcal{N}(s) }[/math] and consequently, if [math]\displaystyle{ \mathcal{B} \to s }[/math] for all [math]\displaystyle{ s \in S }[/math] then [math]\displaystyle{ \mathcal{B} \to S. }[/math] Applying this to [math]\displaystyle{ S := \lim \mathcal{B}, }[/math] this says that if a family [math]\displaystyle{ \mathcal{B} }[/math] has at least one limit point, then it converges to its set of limit points: [math]\displaystyle{ \mathcal{B} \to \lim \mathcal{B}. }[/math]

Basic properties

If [math]\displaystyle{ \mathcal{B} }[/math] converges to a point or subset then the same is true of any family finer than [math]\displaystyle{ \mathcal{B}. }[/math] This has many important consequences. One consequence is that the limit points of a family [math]\displaystyle{ \mathcal{B} }[/math] are the same as the limit points of its upward closure: [math]\displaystyle{ \operatorname{lim}_X \mathcal{B} ~=~ \operatorname{lim}_X \left(\mathcal{B}^{\uparrow X}\right). }[/math] In particular, the limit points of a prefilter are the same as the limit points of the filter that it generates. Another consequence is that if a family converges to a point (or subset) then the same is true of the family's trace/restriction to any given subset of [math]\displaystyle{ X. }[/math] If [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter and [math]\displaystyle{ B \in \mathcal{B} }[/math] then [math]\displaystyle{ \mathcal{B} }[/math] converges to a point (or subset) of [math]\displaystyle{ X }[/math] if and only if this is true of the trace [math]\displaystyle{ \mathcal{B}\big\vert_B. }[/math] If a filter subbase converges to a point or subset then do the filter and the π-system that it generates, although the converse is not guaranteed. For example, the filter subbase [math]\displaystyle{ \{(-\infty, 0], [0, \infty)\} }[/math] does not converge to [math]\displaystyle{ 0 }[/math] in [math]\displaystyle{ X := \R }[/math] although the (principle ultra) filter that it generates does.

Given [math]\displaystyle{ x \in X, }[/math] the following are equivalent for a prefilter [math]\displaystyle{ \mathcal{B}: }[/math]

1. [math]\displaystyle{ \mathcal{B} }[/math] converges to [math]\displaystyle{ x. }[/math]
2. [math]\displaystyle{ \mathcal{B} }[/math] converges to the set [math]\displaystyle{ \{x\}. }[/math]
3. [math]\displaystyle{ \mathcal{B}^{\uparrow X} }[/math] converges to [math]\displaystyle{ x. }[/math]
4. There exists a family equivalent to [math]\displaystyle{ \mathcal{B} }[/math] that converges to [math]\displaystyle{ x. }[/math]

Because subordination is transitive, if [math]\displaystyle{ \mathcal{B} \leq \mathcal{C} \text{ then } \lim {}_{X} \mathcal{B} \subseteq \lim {}_{X} \mathcal{C} }[/math] and moreover, for every [math]\displaystyle{ x \in X, }[/math] both [math]\displaystyle{ \{x\} }[/math] and the maximal/ultrafilter [math]\displaystyle{ \{x\}^{\uparrow X} }[/math] converge to [math]\displaystyle{ x. }[/math] Thus every topological space [math]\displaystyle{ (X, \tau) }[/math] induces a canonical convergence [math]\displaystyle{ \xi \subseteq X \times \operatorname{Filters}(X) }[/math] defined by [math]\displaystyle{ (x, \mathcal{B}) \in \xi \text{ if and only if } x \in \lim {}_{(X, \tau)} \mathcal{B}. }[/math] At the other extreme, the neighborhood filter [math]\displaystyle{ \mathcal{N}(x) }[/math] is the smallest (that is, coarsest) filter on [math]\displaystyle{ X }[/math] that converges to [math]\displaystyle{ x; }[/math] that is, any filter converging to [math]\displaystyle{ x }[/math] must contain [math]\displaystyle{ \mathcal{N}(x) }[/math] as a subset. Said differently, the family of filters that converge to [math]\displaystyle{ x }[/math] consists exactly of those filter on [math]\displaystyle{ X }[/math] that contain [math]\displaystyle{ \mathcal{N}(x) }[/math] as a subset. Consequently, the finer the topology on [math]\displaystyle{ X }[/math] then the fewer prefilters exist that have any limit points in [math]\displaystyle{ X. }[/math]

Cluster points

A family [math]\displaystyle{ \mathcal{B} }[/math] is said to cluster at a point or subset [math]\displaystyle{ x }[/math] of [math]\displaystyle{ X }[/math] if it meshes with the neighborhood filter of [math]\displaystyle{ x; }[/math] that is, if [math]\displaystyle{ \mathcal{B} \# \mathcal{N}(x). }[/math] Explicitly, this means that [math]\displaystyle{ B \cap N \neq \varnothing \text{ for every } B \in \mathcal{B} }[/math] and every neighborhood [math]\displaystyle{ N }[/math] of [math]\displaystyle{ x. }[/math] In particular, a point [math]\displaystyle{ x \in X }[/math] is a cluster point or an accumulation point of a family [math]\displaystyle{ \mathcal{B} }[/math] if [math]\displaystyle{ \mathcal{B} }[/math] meshes with the neighborhood filter at [math]\displaystyle{ x: \ \mathcal{B} \# \mathcal{N}(x). }[/math] The set of all cluster points of [math]\displaystyle{ \mathcal{B} }[/math] is denoted by [math]\displaystyle{ \operatorname{cl}_X \mathcal{B}, }[/math] where the subscript may be dropped if not needed.

In the above definitions, it suffices to check that [math]\displaystyle{ \mathcal{B} }[/math] meshes with some (or equivalently, meshes with every) neighborhood base in [math]\displaystyle{ X }[/math] of [math]\displaystyle{ x \text{ or } S. }[/math] When [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter then the definition of "[math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{N} }[/math] mesh" can be characterized entirely in terms of the subordination preorder [math]\displaystyle{ \,\leq\,. }[/math]

Two equivalent families of sets have the exact same limit points and also the same cluster points. No matter the topology, for every [math]\displaystyle{ x \in X, }[/math] both [math]\displaystyle{ \{x\} }[/math] and the principal ultrafilter [math]\displaystyle{ \{x\}^{\uparrow X} }[/math] cluster at [math]\displaystyle{ x. }[/math] For any [math]\displaystyle{ S \subseteq X, }[/math] if [math]\displaystyle{ \mathcal{B} }[/math] clusters at some [math]\displaystyle{ s \in S }[/math] then [math]\displaystyle{ \mathcal{B} }[/math] clusters at [math]\displaystyle{ S. }[/math] No family clusters at [math]\displaystyle{ S := \varnothing }[/math] and if [math]\displaystyle{ \varnothing \in \mathcal{B} \text{ then } \varnothing = \operatorname{cl} \mathcal{B}. }[/math] If [math]\displaystyle{ \mathcal{B} }[/math] clusters to a point or subset then the same is true of any family coarser than [math]\displaystyle{ \mathcal{B}. }[/math] Consequently, the cluster points of a family [math]\displaystyle{ \mathcal{B} }[/math] are the same as the cluster points of its upward closure: [math]\displaystyle{ \operatorname{cl}_X \mathcal{B} ~=~ \operatorname{cl}_X \left(\mathcal{B}^{\uparrow X}\right). }[/math] In particular, the cluster points of a prefilter are the same as the cluster points of the filter that it generates.

Given [math]\displaystyle{ x \in X, }[/math] the following are equivalent for a prefilter [math]\displaystyle{ \mathcal{B} \text{ on } X }[/math]:

1. [math]\displaystyle{ \mathcal{B} }[/math] clusters at [math]\displaystyle{ x. }[/math]
2. [math]\displaystyle{ \mathcal{B} }[/math] clusters at the set [math]\displaystyle{ \{x\}. }[/math]
3. The family [math]\displaystyle{ \mathcal{B}^{\uparrow X} }[/math] generated by [math]\displaystyle{ \mathcal{B} }[/math] clusters at [math]\displaystyle{ x. }[/math]
4. There exists a family equivalent to [math]\displaystyle{ \mathcal{B} }[/math] that clusters at [math]\displaystyle{ x. }[/math]
5. [math]\displaystyle{ x \in {\textstyle\bigcap\limits_{F \in \mathcal{B}}} \operatorname{cl}_X F. }[/math]
6. [math]\displaystyle{ X \setminus N \not\in \mathcal{B}^{\uparrow X} }[/math] for every neighborhood [math]\displaystyle{ N }[/math] of [math]\displaystyle{ x. }[/math]
   • If [math]\displaystyle{ \mathcal{B} }[/math] is a filter on [math]\displaystyle{ X }[/math] then [math]\displaystyle{ x \in \operatorname{cl}_X \mathcal{B} \text{ if and only if } X \setminus N \not\in \mathcal{B} }[/math] for every neighborhood [math]\displaystyle{ N \text{ of } x. }[/math]
7. There exists a prefilter [math]\displaystyle{ \mathcal{F} }[/math] subordinate to [math]\displaystyle{ \mathcal{B} }[/math] (that is, [math]\displaystyle{ \mathcal{F} \geq \mathcal{B} }[/math]) that converges to [math]\displaystyle{ x. }[/math]
   • This is the filter equivalent of "[math]\displaystyle{ x }[/math] is a cluster point of a sequence if and only if there exists a subsequence converging to [math]\displaystyle{ x. }[/math]
   • In particular, if [math]\displaystyle{ x }[/math] is a cluster point of a prefilter [math]\displaystyle{ \mathcal{B} }[/math] then [math]\displaystyle{ \mathcal{B} (\cap) \mathcal{N}(x) }[/math] is a prefilter subordinate to [math]\displaystyle{ \mathcal{B} }[/math] that converges to [math]\displaystyle{ x. }[/math]

The set [math]\displaystyle{ \operatorname{cl}_X \mathcal{B} }[/math] of all cluster points of a prefilter [math]\displaystyle{ \mathcal{B} }[/math] satisfies [math]\displaystyle{ \operatorname{cl}_X \mathcal{B} = \bigcap_{B \in \mathcal{B}} \operatorname{cl}_X B. }[/math] Consequently, the set [math]\displaystyle{ \operatorname{cl}_X \mathcal{B} }[/math] of all cluster points of any prefilter [math]\displaystyle{ \mathcal{B} }[/math] is a closed subset of [math]\displaystyle{ X. }[/math] This also justifies the notation [math]\displaystyle{ \operatorname{cl}_X \mathcal{B} }[/math] for the set of cluster points. In particular, if [math]\displaystyle{ K \subseteq X }[/math] is non-empty (so that [math]\displaystyle{ \mathcal{B} := \{K\} }[/math] is a prefilter) then [math]\displaystyle{ \operatorname{cl}_X \{K\} = \operatorname{cl}_X K }[/math] since both sides are equal to [math]\displaystyle{ {\textstyle\bigcap\limits_{B \in \mathcal{B}}} \operatorname{cl}_X B. }[/math]

Properties and relationships

Just like sequences and nets, it is possible for a prefilter on a topological space of infinite cardinality to not have any cluster points or limit points.

If [math]\displaystyle{ x }[/math] is a limit point of [math]\displaystyle{ \mathcal{B} }[/math] then [math]\displaystyle{ x }[/math] is necessarily a limit point of any family [math]\displaystyle{ \mathcal{C} }[/math] finer than [math]\displaystyle{ \mathcal{B} }[/math] (that is, if [math]\displaystyle{ \mathcal{N}(x) \leq \mathcal{B} \text{ and } \mathcal{B} \leq \mathcal{C} }[/math] then [math]\displaystyle{ \mathcal{N}(x) \leq \mathcal{C} }[/math]). In contrast, if [math]\displaystyle{ x }[/math] is a cluster point of [math]\displaystyle{ \mathcal{B} }[/math] then [math]\displaystyle{ x }[/math] is necessarily a cluster point of any family [math]\displaystyle{ \mathcal{C} }[/math] coarser than [math]\displaystyle{ \mathcal{B} }[/math] (that is, if [math]\displaystyle{ \mathcal{N}(x) \text{ and } \mathcal{B} }[/math] mesh and [math]\displaystyle{ \mathcal{C} \leq \mathcal{B} }[/math] then [math]\displaystyle{ \mathcal{N}(x) \text{ and } \mathcal{C} }[/math] mesh).

Equivalent families and subordination

Any two equivalent families [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{C} }[/math] can be used interchangeably in the definitions of "limit of" and "cluster at" because their equivalency guarantees that [math]\displaystyle{ \mathcal{N} \leq \mathcal{B} }[/math] if and only if [math]\displaystyle{ \mathcal{N} \leq \mathcal{C}, }[/math] and also that [math]\displaystyle{ \mathcal{N} \# \mathcal{B} }[/math] if and only if [math]\displaystyle{ \mathcal{N} \# \mathcal{C}. }[/math] In essence, the preorder [math]\displaystyle{ \,\leq\, }[/math] is incapable of distinguishing between equivalent families. Given two prefilters, whether or not they mesh can be characterized entirely in terms of subordination. Thus the two most fundamental concepts related to (pre)filters to Topology (that is, limit and cluster points) can both be defined entirely in terms of the subordination relation. This is why the preorder [math]\displaystyle{ \,\leq\, }[/math] is of such great importance in applying (pre)filters to Topology.

Limit and cluster point relationships and sufficient conditions

Every limit point of a non-degenerate family [math]\displaystyle{ \mathcal{B} }[/math] is also a cluster point; in symbols: [math]\displaystyle{ \operatorname{lim}_X \mathcal{B} ~\subseteq~ \operatorname{cl}_X \mathcal{B}. }[/math] This is because if [math]\displaystyle{ x }[/math] is a limit point of [math]\displaystyle{ \mathcal{B} }[/math] then [math]\displaystyle{ \mathcal{N}(x) \text{ and } \mathcal{B} }[/math] mesh, which makes [math]\displaystyle{ x }[/math] a cluster point of [math]\displaystyle{ \mathcal{B}. }[/math] But in general, a cluster point need not be a limit point. For instance, every point in any given non-empty subset [math]\displaystyle{ K \subseteq X }[/math] is a cluster point of the principle prefilter [math]\displaystyle{ \mathcal{B} := \{K\} }[/math] (no matter what topology is on [math]\displaystyle{ X }[/math]) but if [math]\displaystyle{ X }[/math] is Hausdorff and [math]\displaystyle{ K }[/math] has more than one point then this prefilter has no limit points; the same is true of the filter [math]\displaystyle{ \{K\}^{\uparrow X} }[/math] that this prefilter generates.

However, every cluster point of an ultra prefilter is a limit point. Consequently, the limit points of an ultra prefilter [math]\displaystyle{ \mathcal{B} }[/math] are the same as its cluster points: [math]\displaystyle{ \operatorname{lim}_X \mathcal{B} = \operatorname{cl}_X \mathcal{B}; }[/math] that is to say, a given point is a cluster point of an ultra prefilter [math]\displaystyle{ \mathcal{B} }[/math] if and only if [math]\displaystyle{ \mathcal{B} }[/math] converges to that point. Although a cluster point of a filter need not be a limit point, there will always exist a finer filter that does converge to it; in particular, if [math]\displaystyle{ \mathcal{B} }[/math] clusters at [math]\displaystyle{ x }[/math] then [math]\displaystyle{ \mathcal{B} \,(\cap)\, \mathcal{N}(x) = \{B \cap N : B \in \mathcal{B}, N \in \mathcal{N}(x)\} }[/math] is a filter subbase whose generated filter converges to [math]\displaystyle{ x. }[/math]

If [math]\displaystyle{ \varnothing \neq \mathcal{B} \subseteq \wp(X) \text{ and } \mathcal{S} \geq \mathcal{B} }[/math] is a filter subbase such that [math]\displaystyle{ \mathcal{S} \to x \text{ in } X }[/math] then [math]\displaystyle{ x \in \operatorname{cl}_X \mathcal{B}. }[/math] In particular, any limit point of a filter subbase subordinate to [math]\displaystyle{ \mathcal{B} \neq \varnothing }[/math] is necessarily also a cluster point of [math]\displaystyle{ \mathcal{B}. }[/math] If [math]\displaystyle{ x }[/math] is a cluster point of a prefilter [math]\displaystyle{ \mathcal{B} }[/math] then [math]\displaystyle{ \mathcal{B} (\cap) \mathcal{N}(x) }[/math] is a prefilter subordinate to [math]\displaystyle{ \mathcal{B} }[/math] that converges to [math]\displaystyle{ x \text{ in } X. }[/math]

If [math]\displaystyle{ S \subseteq X }[/math] and if [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter on [math]\displaystyle{ S }[/math] then every cluster point of [math]\displaystyle{ \mathcal{B} \text{ in } X }[/math] belongs to [math]\displaystyle{ \operatorname{cl}_X S }[/math] and any point in [math]\displaystyle{ \operatorname{cl}_X S }[/math] is a limit point of a filter on [math]\displaystyle{ S. }[/math]

Primitive sets

A subset [math]\displaystyle{ P \subseteq X }[/math] is called primitive if it is the set of limit points of some ultrafilter (or equivalently, some ultra prefilter). That is, if there exists an ultrafilter [math]\displaystyle{ \mathcal{B} \text{ on } X }[/math] such that [math]\displaystyle{ P }[/math] is equal to [math]\displaystyle{ \operatorname{lim}_X \mathcal{B}, }[/math] which recall denotes the set of limit points of [math]\displaystyle{ \mathcal{B} \text{ in } X. }[/math] Since limit points are the same as cluster points for ultra prefilters, a subset is primitive if and only if it is equal to the set [math]\displaystyle{ \operatorname{cl}_X \mathcal{B} }[/math] of cluster points of some ultra prefilter [math]\displaystyle{ \mathcal{B}. }[/math] For example, every closed singleton subset is primitive. The image of a primitive subset of [math]\displaystyle{ X }[/math] under a continuous map [math]\displaystyle{ f : X \to Y }[/math] is contained in a primitive subset of [math]\displaystyle{ Y. }[/math]

Assume that [math]\displaystyle{ P, Q \subseteq X }[/math] are two primitive subset of [math]\displaystyle{ X. }[/math] If [math]\displaystyle{ U }[/math] is an open subset of [math]\displaystyle{ X }[/math] that intersects [math]\displaystyle{ P }[/math] then [math]\displaystyle{ U \in \mathcal{B} }[/math] for any ultrafilter [math]\displaystyle{ \mathcal{B} \text{ on } X }[/math] such that [math]\displaystyle{ P = \operatorname{lim}_X \mathcal{B}. }[/math] In addition, if [math]\displaystyle{ P \text{ and } Q }[/math] are distinct then there exists some [math]\displaystyle{ S \subseteq X }[/math] and some ultrafilters [math]\displaystyle{ \mathcal{B}_P \text{ and } \mathcal{B}_Q \text{ on } X }[/math] such that [math]\displaystyle{ P = \operatorname{lim}_X \mathcal{B}_P, Q = \operatorname{lim}_X \mathcal{B}_Q, S \in \mathcal{B}_P, }[/math] and [math]\displaystyle{ X \setminus S \in \mathcal{B}_Q. }[/math]

Other results

If [math]\displaystyle{ X }[/math] is a complete lattice then:

• The limit inferior of [math]\displaystyle{ B }[/math] is the infimum of the set of all cluster points of [math]\displaystyle{ B. }[/math]
• The limit superior of [math]\displaystyle{ B }[/math] is the supremum of the set of all cluster points of [math]\displaystyle{ B. }[/math]
• [math]\displaystyle{ B }[/math] is a convergent prefilter if and only if its limit inferior and limit superior agree; in this case, the value on which they agree is the limit of the prefilter.

Limits of functions defined as limits of prefilters

Suppose [math]\displaystyle{ f : X \to Y }[/math] is a map from a set into a topological space [math]\displaystyle{ Y, }[/math] [math]\displaystyle{ \mathcal{B} \subseteq \wp(X), }[/math] and [math]\displaystyle{ y \in Y. }[/math] If [math]\displaystyle{ y }[/math] is a limit point (respectively, a cluster point) of [math]\displaystyle{ f(\mathcal{B}) \text{ in } Y }[/math] then [math]\displaystyle{ y }[/math] is called a limit point or limit (respectively, a cluster point) of [math]\displaystyle{ f }[/math] with respect to [math]\displaystyle{ \mathcal{B}. }[/math] Explicitly, [math]\displaystyle{ y }[/math] is a limit of [math]\displaystyle{ f }[/math] with respect to [math]\displaystyle{ \mathcal{B} }[/math] if and only if [math]\displaystyle{ \mathcal{N}(y) \leq f(\mathcal{B}), }[/math] which can be written as [math]\displaystyle{ f(\mathcal{B}) \to y \text{ or } \lim f(\mathcal{B}) \to y \text{ in } Y }[/math] (by definition of this notation) and stated as [math]\displaystyle{ f }[/math] tend to [math]\displaystyle{ y }[/math] along [math]\displaystyle{ \mathcal{B}. }[/math] If the limit [math]\displaystyle{ y }[/math] is unique then the arrow [math]\displaystyle{ \to }[/math] may be replaced with an equals sign [math]\displaystyle{ =. }[/math] The neighborhood filter [math]\displaystyle{ \mathcal{N}(y) }[/math] can be replaced with any family equivalent to it and the same is true of [math]\displaystyle{ \mathcal{B}. }[/math]

The definition of a convergent net is a special case of the above definition of a limit of a function. Specifically, if [math]\displaystyle{ x \in X \text{ and } \chi : (I, \leq) \to X }[/math] is a net then [math]\displaystyle{ \chi \to x \text{ in } X \quad \text{ if and only if } \quad \chi(\operatorname{Tails}(I, \leq)) \to x \text{ in } X, }[/math] where the left hand side states that [math]\displaystyle{ x }[/math] is a limit of the net [math]\displaystyle{ \chi }[/math] while the right hand side states that [math]\displaystyle{ x }[/math] is a limit of the function [math]\displaystyle{ \chi }[/math] with respect to [math]\displaystyle{ \mathcal{B} := \operatorname{Tails}(I, \leq) }[/math] (as just defined above).

The table below shows how various types of limits encountered in analysis and topology can be defined in terms of the convergence of images (under [math]\displaystyle{ f }[/math]) of particular prefilters on the domain [math]\displaystyle{ X. }[/math] This shows that prefilters provide a general framework into which many of the various definitions of limits fit. The limits in the left–most column are defined in their usual way with their obvious definitions.

Throughout, let [math]\displaystyle{ f : X \to Y }[/math] be a map between topological spaces, [math]\displaystyle{ x_0 \in X, \text{ and } y \in Y. }[/math] If [math]\displaystyle{ Y }[/math] is Hausdorff then all arrows "[math]\displaystyle{ \to y }[/math]" in the table may be replaced with equal signs "[math]\displaystyle{ = y }[/math]" and "[math]\displaystyle{ \lim f(\mathcal{B}) \to y }[/math]" may be replaced with "[math]\displaystyle{ \lim f(\mathcal{B}) = y }[/math]".

Type of limit	Template:Nowarp	Definition in terms of prefilters	Assumptions
[math]\displaystyle{ \lim_{x \to x_0} f(x) \to y }[/math]	⇔	[math]\displaystyle{ f(\mathcal{B}) \to y \text{ where } \mathcal{B}\,:=\,\mathcal{N}\left(x_0\right) }[/math]
[math]\displaystyle{ \lim_{\stackrel{x \to x_0}{x \neq x_0}} f(x) \to y }[/math]	⇔	[math]\displaystyle{ f(\mathcal{B}) \to y \text{ where } \mathcal{B}\,:=\,\left\{N \setminus \left\{x_0\right\} : N \in \mathcal{N}\left(x_0\right)\right\} }[/math]
[math]\displaystyle{ \lim_{\stackrel{x \to x_0}{x \in S}} f(x) \to y }[/math] or [math]\displaystyle{ \lim_{x \to x_0} f\big\vert_S(x) \to y }[/math]	⇔	[math]\displaystyle{ f(\mathcal{B}) \to y \text{ where } \mathcal{B}\,:=\,\mathcal{N}\left(x_0\right)\big\vert_S\,:=\,\left\{N \cap S : N \in \mathcal{N}\left(x_0\right)\right\} }[/math]	[math]\displaystyle{ S \subseteq X \text{ and } x_0 \in \operatorname{cl}_X S }[/math]
[math]\displaystyle{ \lim_{\stackrel{x \to x_0}{x \neq x_0}} f(x) \to y }[/math]	⇔	[math]\displaystyle{ f(\mathcal{B}) \to y \text{ where } \mathcal{B}\,:=\,\left\{\left(x_0 - r, x_0\right) \cup \left(x_0, x_0 + r\right) : 0 \lt r \in \R\right\} }[/math]	[math]\displaystyle{ x_0 \in X = \R }[/math]
[math]\displaystyle{ \lim_{\stackrel{x \to x_0}{x \lt x_0}} f(x) \to y }[/math]	⇔	[math]\displaystyle{ f(\mathcal{B}) \to y \text{ where } \mathcal{B}\,:=\,\left\{\left(x, x_0\right) : x \lt x_0\right\} }[/math]	[math]\displaystyle{ x_0 \in X = \R }[/math]
[math]\displaystyle{ \lim_{\stackrel{x \to x_0}{x \leq x_0}} f(x) \to y }[/math]	⇔	[math]\displaystyle{ f(\mathcal{B}) \to y \text{ where } \mathcal{B}\,:=\,\left\{\left(x, x_0\right] : x \lt x_0\right\} }[/math]	[math]\displaystyle{ x_0 \in X = \R }[/math]
[math]\displaystyle{ \lim_{\stackrel{x \to x_0}{x \gt x_0}} f(x) \to y }[/math]	⇔	[math]\displaystyle{ f(\mathcal{B}) \to y \text{ where } \mathcal{B}\,:=\,\left\{\left(x_0, x\right) : x_0 \lt x\right\} }[/math]	[math]\displaystyle{ x_0 \in X = \R }[/math]
[math]\displaystyle{ \lim_{\stackrel{x \to x_0}{x \geq x_0}} f(x) \to y }[/math]	⇔	[math]\displaystyle{ f(\mathcal{B}) \to y \text{ where } \mathcal{B}\,:=\,\left\{\left[x_0, x\right) : x_0 \leq x\right\} }[/math]	[math]\displaystyle{ x_0 \in X = \R }[/math]
[math]\displaystyle{ \lim_{n \to \infty} f(n) \to y }[/math]	⇔	[math]\displaystyle{ f(\mathcal{B}) \to y \text{ where } \mathcal{B}\,:=\,\{\{n, n + 1, \ldots\} ~:~ n \in \N\}\} }[/math]	[math]\displaystyle{ X = \N \text{ so } f : \N \to Y }[/math] is a sequence in [math]\displaystyle{ Y }[/math]
[math]\displaystyle{ \lim_{x \to \infty} f(x) \to y }[/math]	⇔	[math]\displaystyle{ f(\mathcal{B}) \to y \text{ where } \mathcal{B}\,:=\,(\R, \infty) \,:=\,\{(x, \infty) : x \in \R\} }[/math]	[math]\displaystyle{ X = \R }[/math]
[math]\displaystyle{ \lim_{x \to -\infty} f(x) \to y }[/math]	⇔	[math]\displaystyle{ f(\mathcal{B}) \to y \text{ where } \mathcal{B}\,:=\,(-\infty,\R) \,:=\,\{(-\infty, x) : x \in \R\} }[/math]	[math]\displaystyle{ X = \R }[/math]
[math]\displaystyle{ \lim_{|x| \to \infty} f(x) \to y }[/math]	⇔	[math]\displaystyle{ f(\mathcal{B}) \to y \text{ where } \mathcal{B}\,:=\,\{X \cap [(-\infty, x) \cup (x, \infty)] : x \in \R\} }[/math]	[math]\displaystyle{ X = \R \text{ or } X = \Z }[/math] for a double-ended sequence
[math]\displaystyle{ \lim_{\|x\| \to \infty} f(x) \to y }[/math]	⇔	[math]\displaystyle{ f(\mathcal{B}) \to y \text{ where } \mathcal{B}\,:=\,\{\{x \in X : \|x\| \gt r\} ~:~ 0 \lt r \in \R\} }[/math]	[math]\displaystyle{ (X, \|\cdot\|) \text{ is} }[/math] a seminormed space; for example, a Banach space [math]\displaystyle{ \text{like } X = \Complex }[/math]

By defining different prefilters, many other notions of limits can be defined; for example, [math]\displaystyle{ \lim_{\stackrel{|x| \to |x_0|}{|x| \neq |x_0|}} f(x) \to y. }[/math]

Divergence to infinity

Divergence of a real-valued function to infinity can be defined/characterized by using the prefilters [math]\displaystyle{ (\R, \infty) := \{(r, \infty) : r \in \R\} ~~ \text{ and } ~~ (-\infty, \R) := \{(-\infty, r) : r \in \R\}, }[/math] where [math]\displaystyle{ f \to \infty }[/math] along [math]\displaystyle{ \mathcal{B} }[/math] if and only if [math]\displaystyle{ (\R, \infty) \leq f(\mathcal{B}) }[/math] and similarly, [math]\displaystyle{ f \to -\infty }[/math] along [math]\displaystyle{ \mathcal{B} }[/math] if and only if [math]\displaystyle{ (-\infty, \R) \leq f(\mathcal{B}). }[/math] The family [math]\displaystyle{ (\R, \infty) }[/math] can be replaced by any family equivalent to it, such as [math]\displaystyle{ [\R, \infty) := \{[r, \infty) : r \in \R\} }[/math] for instance (in real analysis, this would correspond to replacing the strict inequality "[math]\displaystyle{ f(x) \gt r }[/math]" in the definition with "[math]\displaystyle{ f(x) \geq r }[/math]"), and the same is true of [math]\displaystyle{ \mathcal{B} }[/math] and [math]\displaystyle{ (-\infty, \R). }[/math]

So for example, if [math]\displaystyle{ \mathcal{B}\,:=\,\mathcal{N}\left(x_0\right) }[/math] then [math]\displaystyle{ \lim_{x \to x_0} f(x) \to \infty }[/math] if and only if [math]\displaystyle{ (\R, \infty) \leq f(\mathcal{B}) }[/math] holds. Similarly, [math]\displaystyle{ \lim_{x \to x_0} f(x) \to - \infty }[/math] if and only if [math]\displaystyle{ (-\infty, \R) \leq f\left(\mathcal{N}\left(x_0\right)\right), }[/math] or equivalently, if and only if [math]\displaystyle{ (-\infty, \R] \leq f\left(\mathcal{N}\left(x_0\right)\right). }[/math]

More generally, if [math]\displaystyle{ f }[/math] is valued in [math]\displaystyle{ Y = \R^n \text{ or } Y = \Complex^n }[/math] (or some other seminormed vector space) and if [math]\displaystyle{ B_{\geq r} := \{y \in Y : |y| \geq r\} = Y \setminus B_{\lt r} }[/math] then [math]\displaystyle{ \lim_{x \to x_0} |f(x)| \to \infty }[/math] if and only if [math]\displaystyle{ B_{\geq \R} \leq f\left(\mathcal{N}\left(x_0\right)\right) }[/math] holds, where [math]\displaystyle{ B_{\geq \R} := \left\{B_{\geq r} : r \in \R\right\}. }[/math]

Filters and nets

This section will describe the relationships between prefilters and nets in great detail because of how important these details are applying filters to topology − particularly in switching from utilizing nets to utilizing filters and vice verse.

Nets to prefilters

In the definitions below, the first statement is the standard definition of a limit point of a net (respectively, a cluster point of a net) and it is gradually reworded until the corresponding filter concept is reached.

A net [math]\displaystyle{ x_\bull = \left(x_i\right)_{i \in I} }[/math] is said to converge in [math]\displaystyle{ (X, \tau) }[/math] to a point [math]\displaystyle{ x \in X, }[/math] written [math]\displaystyle{ x_\bull \to x \text{ in } X, }[/math] and [math]\displaystyle{ x }[/math] is called a limit or limit point of [math]\displaystyle{ x_\bull, }[/math] if any of the following equivalent conditions hold:

1. Definition: For every [math]\displaystyle{ N \in \mathcal{N}_{\tau}(x), }[/math] there exists some [math]\displaystyle{ i \in I }[/math] such that if [math]\displaystyle{ i \leq j \in I \text{ then } x_j \in N. }[/math]
2. For every [math]\displaystyle{ N \in \mathcal{N}_{\tau}(x), }[/math] there exists some [math]\displaystyle{ i \in I }[/math] such that the tail of [math]\displaystyle{ x_\bull }[/math] starting at [math]\displaystyle{ i }[/math] is contained in [math]\displaystyle{ N }[/math] (that is, such that [math]\displaystyle{ x_{\geq i} \subseteq N }[/math]).
3. For every [math]\displaystyle{ N \in \mathcal{N}_{\tau}(x), }[/math] there exists some [math]\displaystyle{ B \in \operatorname{Tails}\left(x_\bull\right) }[/math] such that [math]\displaystyle{ B \subseteq N. }[/math]
4. [math]\displaystyle{ \mathcal{N}_{\tau}(x) \leq \operatorname{Tails}\left(x_\bull\right). }[/math]
5. [math]\displaystyle{ \operatorname{Tails}\left(x_\bull\right) \to x \text{ in } X; }[/math] that is, the prefilter [math]\displaystyle{ \operatorname{Tails}\left(x_\bull\right) }[/math] converges to [math]\displaystyle{ x. }[/math]

As usual, [math]\displaystyle{ \lim x_\bull = x }[/math] is defined to mean that [math]\displaystyle{ x_\bull \to x }[/math] and [math]\displaystyle{ x }[/math] is the only limit point of [math]\displaystyle{ x_\bull; }[/math] that is, if also [math]\displaystyle{ x_\bull \to z \text{ then } z = x. }[/math]

A point [math]\displaystyle{ x \in X }[/math] is called a cluster or accumulation point of a net [math]\displaystyle{ x_\bull = \left(x_i\right)_{i \in I} \text{ in } (X, \tau) }[/math] if any of the following equivalent conditions hold:

1. Definition: For every [math]\displaystyle{ N \in \mathcal{N}_{\tau}(x) }[/math] and every [math]\displaystyle{ i \in I, }[/math] there exists some [math]\displaystyle{ i \leq j \in I }[/math] such that [math]\displaystyle{ x_j \in N. }[/math]
2. For every [math]\displaystyle{ N \in \mathcal{N}_{\tau}(x) }[/math] and every [math]\displaystyle{ i \in I, }[/math] the tail of [math]\displaystyle{ x_\bull }[/math] starting at [math]\displaystyle{ i }[/math] intersects [math]\displaystyle{ N }[/math] (that is, [math]\displaystyle{ x_{\geq i} \cap N \neq \varnothing }[/math]).
3. For every [math]\displaystyle{ N \in \mathcal{N}_{\tau}(x) }[/math] and every [math]\displaystyle{ B \in \operatorname{Tails}\left(x_\bull\right), B \cap N \neq \varnothing. }[/math]
4. [math]\displaystyle{ \mathcal{N}_{\tau}(x) \text{ and } \operatorname{Tails}\left(x_\bull\right) }[/math] mesh (by definition of "mesh").
5. [math]\displaystyle{ x }[/math] is a cluster point of [math]\displaystyle{ \operatorname{Tails}\left(x_\bull\right). }[/math]

If [math]\displaystyle{ f : X \to Y }[/math] is a map and [math]\displaystyle{ x_\bull }[/math] is a net in [math]\displaystyle{ X }[/math] then [math]\displaystyle{ \operatorname{Tails}\left(f\left(x_\bull\right)\right) = f\left(\operatorname{Tails}\left(x_\bull\right)\right). }[/math]

Prefilters to nets

A pointed set is a pair [math]\displaystyle{ (S, s) }[/math] consisting of a non–empty set [math]\displaystyle{ S }[/math] and an element [math]\displaystyle{ s \in S. }[/math] For any family [math]\displaystyle{ \mathcal{B}, }[/math] let [math]\displaystyle{ \operatorname{PointedSets}(\mathcal{B}) := \{(B, b) ~:~ B \in \mathcal{B} \text{ and } b \in B\}. }[/math]

Define a canonical preorder [math]\displaystyle{ \,\leq\, }[/math] on pointed sets by declaring [math]\displaystyle{ (R, r) \leq (S, s) \quad \text{ if and only if } \quad R \supseteq S. }[/math]

There is a canonical map [math]\displaystyle{ \operatorname{Point}_{\mathcal{B}} ~:~ \operatorname{PointedSets}(\mathcal{B}) \to X }[/math] defined by [math]\displaystyle{ (B, b) \mapsto b. }[/math] If [math]\displaystyle{ i_0 = \left(B_0, b_0\right) \in \operatorname{PointedSets}(\mathcal{B}) }[/math] then the tail of the assignment [math]\displaystyle{ \operatorname{Point}_{\mathcal{B}} }[/math] starting at [math]\displaystyle{ i_0 }[/math] is [math]\displaystyle{ \left\{c ~:~ (C, c) \in \operatorname{PointedSets}(\mathcal{B}) \text{ and } \left(B_0, b_0\right) \leq (C, c)\right\} = B_0. }[/math]

Although [math]\displaystyle{ (\operatorname{PointedSets}(\mathcal{B}), \leq) }[/math] is not, in general, a partially ordered set, it is a directed set if (and only if) [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter. So the most immediate choice for the definition of "the net in [math]\displaystyle{ X }[/math] induced by a prefilter [math]\displaystyle{ \mathcal{B} }[/math]" is the assignment [math]\displaystyle{ (B, b) \mapsto b }[/math] from [math]\displaystyle{ \operatorname{PointedSets}(\mathcal{B}) }[/math] into [math]\displaystyle{ X. }[/math]

If [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter on [math]\displaystyle{ X }[/math] then the net associated with [math]\displaystyle{ \mathcal{B} }[/math] is the map

[math]\displaystyle{ \begin{alignat}{4} \operatorname{Net}_{\mathcal{B}} :\;&& (\operatorname{PointedSets}(\mathcal{B}), \leq) &&\,\to \;& X \\ && (B, b) &&\,\mapsto\;& b \\ \end{alignat} }[/math] that is, [math]\displaystyle{ \operatorname{Net}_{\mathcal{B}}(B, b) := b. }[/math]

If [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter on [math]\displaystyle{ X \text{ then } \operatorname{Net}_{\mathcal{B}} }[/math] is a net in [math]\displaystyle{ X }[/math] and the prefilter associated with [math]\displaystyle{ \operatorname{Net}_{\mathcal{B}} }[/math] is [math]\displaystyle{ \mathcal{B} }[/math]; that is: [math]\displaystyle{ \operatorname{Tails}\left(\operatorname{Net}_{\mathcal{B}}\right) = \mathcal{B}. }[/math] This would not necessarily be true had [math]\displaystyle{ \operatorname{Net}_{\mathcal{B}} }[/math] been defined on a proper subset of [math]\displaystyle{ \operatorname{PointedSets}(\mathcal{B}). }[/math]

If [math]\displaystyle{ x_\bull }[/math] is a net in [math]\displaystyle{ X }[/math] then it is not in general true that [math]\displaystyle{ \operatorname{Net}_{\operatorname{Tails}\left(x_\bull\right)} }[/math] is equal to [math]\displaystyle{ x_\bull }[/math] because, for example, the domain of [math]\displaystyle{ x_\bull }[/math] may be of a completely different cardinality than that of [math]\displaystyle{ \operatorname{Net}_{\operatorname{Tails}\left(x_\bull\right)} }[/math] (since unlike the domain of [math]\displaystyle{ \operatorname{Net}_{\operatorname{Tails}\left(x_\bull\right)}, }[/math] the domain of an arbitrary net in [math]\displaystyle{ X }[/math] could have any cardinality).

Proposition — If [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter on [math]\displaystyle{ X }[/math] and [math]\displaystyle{ x \in X }[/math] then

1. [math]\displaystyle{ \mathcal{B} \to x \text{ if and only if } \operatorname{Net}_{\mathcal{B}} \to x. }[/math]
2. [math]\displaystyle{ x }[/math] is a cluster point of [math]\displaystyle{ \mathcal{B} }[/math] if and only if [math]\displaystyle{ x }[/math] is a cluster point of [math]\displaystyle{ \operatorname{Net}_{\mathcal{B}}. }[/math]

Proof

Recall that [math]\displaystyle{ \mathcal{B} = \operatorname{Tails}\left(\operatorname{Net}_{\mathcal{B}}\right) }[/math] and that if [math]\displaystyle{ x_\bull }[/math] is a net in [math]\displaystyle{ X }[/math] then (1) [math]\displaystyle{ x_\bull \to x \text{ if and only if } \operatorname{Tails}\left(x_\bull\right) \to x, }[/math] and (2) [math]\displaystyle{ x }[/math] is a cluster point of [math]\displaystyle{ x_\bull }[/math] if and only if [math]\displaystyle{ x }[/math] is a cluster point of [math]\displaystyle{ \operatorname{Tails}\left(x_\bull\right). }[/math] By using [math]\displaystyle{ x_\bull := \operatorname{Net}_{\mathcal{B}} \text{ and } \mathcal{B} = \operatorname{Tails}\left(\operatorname{Net}_{\mathcal{B}}\right), }[/math] it follows that [math]\displaystyle{ \mathcal{B} \to x \quad \text{ if and only if } \quad \operatorname{Tails}\left(\operatorname{Net}_{\mathcal{B}}\right) \to x \quad \text{ if and only if } \quad \operatorname{Net}_{\mathcal{B}} \to x. }[/math] It also follows that [math]\displaystyle{ x }[/math] is a cluster point of [math]\displaystyle{ \mathcal{B} }[/math] if and only if [math]\displaystyle{ x }[/math] is a cluster point of [math]\displaystyle{ \operatorname{Tails}\left(\operatorname{Net}_{\mathcal{B}}\right) }[/math] if and only if [math]\displaystyle{ x }[/math] is a cluster point of [math]\displaystyle{ \operatorname{Net}_{\mathcal{B}}. }[/math]

Partially ordered net

The domain of the canonical net [math]\displaystyle{ \operatorname{Net}_{\mathcal{B}} }[/math] is in general not partially ordered. However, in 1955 Bruns and Schmidt discovered a construction (detailed here: Filter (set theory)) that allows for the canonical net to have a domain that is both partially ordered and directed; this was independently rediscovered by Albert Wilansky in 1970. Because the tails of this partially ordered net are identical to the tails of [math]\displaystyle{ \operatorname{Net}_{\mathcal{B}} }[/math] (since both are equal to the prefilter [math]\displaystyle{ \mathcal{B} }[/math]), there is typically nothing lost by assuming that the domain of the net associated with a prefilter is both directed and partially ordered. If can further be assumed that the partially ordered domain is also a dense order.

Subordinate filters and subnets

The notion of "[math]\displaystyle{ \mathcal{B} }[/math] is subordinate to [math]\displaystyle{ \mathcal{C} }[/math]" (written [math]\displaystyle{ \mathcal{B} \vdash \mathcal{C} }[/math]) is for filters and prefilters what "[math]\displaystyle{ x_{n_\bull} = \left(x_{n_i}\right)_{i=1}^{\infty} }[/math] is a subsequence of [math]\displaystyle{ x_\bull = \left(x_i\right)_{i=1}^{\infty} }[/math]" is for sequences. For example, if [math]\displaystyle{ \operatorname{Tails}\left(x_\bull\right) = \left\{x_{\geq i} : i \in \N\right\} }[/math] denotes the set of tails of [math]\displaystyle{ x_\bull }[/math] and if [math]\displaystyle{ \operatorname{Tails}\left(x_{n_\bull}\right) = \left\{x_{n_{\geq i}} : i \in \N\right\} }[/math] denotes the set of tails of the subsequence [math]\displaystyle{ x_{n_\bull} }[/math] (where [math]\displaystyle{ x_{n_{\geq i}} := \left\{x_{n_j} ~:~ j \geq i \text{ and } j \in \N\right\} }[/math]) then [math]\displaystyle{ \operatorname{Tails}\left(x_{n_\bull}\right) ~\vdash~ \operatorname{Tails}\left(x_\bull\right) }[/math] (which by definition means [math]\displaystyle{ \operatorname{Tails}\left(x_\bull\right) \leq \operatorname{Tails}\left(x_{n_\bull}\right) }[/math]) is true but [math]\displaystyle{ \operatorname{Tails}\left(x_\bull\right) ~\vdash~ \operatorname{Tails}\left(x_{n_\bull}\right) }[/math] is in general false. If [math]\displaystyle{ x_\bull = \left(x_i\right)_{i \in I} }[/math] is a net in a topological space [math]\displaystyle{ X }[/math] and if [math]\displaystyle{ \mathcal{N}(x) }[/math] is the neighborhood filter at a point [math]\displaystyle{ x \in X, }[/math] then [math]\displaystyle{ x_\bull \to x \text{ if and only if } \mathcal{N}(x) \leq \operatorname{Tails}\left(x_\bull\right). }[/math]

If [math]\displaystyle{ f : X \to Y }[/math] is an surjective open map, [math]\displaystyle{ x \in X, }[/math] and [math]\displaystyle{ \mathcal{C} }[/math] is a prefilter on [math]\displaystyle{ Y }[/math] that converges to [math]\displaystyle{ f(x), }[/math] then there exist a prefilter [math]\displaystyle{ \mathcal{B} }[/math] on [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ \mathcal{B} \to x }[/math] and [math]\displaystyle{ f(\mathcal{B}) }[/math] is equivalent to [math]\displaystyle{ \mathcal{C} }[/math] (that is, [math]\displaystyle{ \mathcal{C} \leq f(\mathcal{B}) \leq \mathcal{C} }[/math]).

Subordination analogs of results involving subsequences

The following results are the prefilter analogs of statements involving subsequences. The condition "[math]\displaystyle{ \mathcal{C} \geq \mathcal{B}, }[/math]" which is also written [math]\displaystyle{ \mathcal{C} \vdash \mathcal{B}, }[/math] is the analog of "[math]\displaystyle{ \mathcal{C} }[/math] is a subsequence of [math]\displaystyle{ \mathcal{B}. }[/math]" So "finer than" and "subordinate to" is the prefilter analog of "subsequence of." Some people prefer saying "subordinate to" instead of "finer than" because it is more reminiscent of "subsequence of."

Proposition — Let [math]\displaystyle{ \mathcal{B} }[/math] be a prefilter on [math]\displaystyle{ X }[/math] and let [math]\displaystyle{ x \in X. }[/math]

1. Suppose [math]\displaystyle{ \mathcal{C} }[/math] is a prefilter such that [math]\displaystyle{ \mathcal{C} \geq \mathcal{B}. }[/math]
   1. If [math]\displaystyle{ \mathcal{B} \to x \text{ then } \mathcal{C} \to x. }[/math]
      • This is the analog of "if a sequence converges to [math]\displaystyle{ x }[/math] then so does every subsequence."
   2. If [math]\displaystyle{ x }[/math] is a cluster point of [math]\displaystyle{ \mathcal{C} }[/math] then [math]\displaystyle{ x }[/math] is a cluster point of [math]\displaystyle{ \mathcal{B}. }[/math]
      • This is the analog of "if [math]\displaystyle{ x }[/math] is a cluster point of some subsequence, then [math]\displaystyle{ x }[/math] is a cluster point of the original sequence."
2. [math]\displaystyle{ \mathcal{B} \to x }[/math] if and only if for any finer prefilter [math]\displaystyle{ \mathcal{C} \geq \mathcal{B} }[/math] there exists some even more fine prefilter [math]\displaystyle{ \mathcal{F} \geq \mathcal{C} }[/math] such that [math]\displaystyle{ \mathcal{F} \to x. }[/math]
   • This is the analog of "a sequence converges to [math]\displaystyle{ x }[/math] if and only if every subsequence has a sub–subsequence that converges to [math]\displaystyle{ x. }[/math]"
3. [math]\displaystyle{ x }[/math] is a cluster point of [math]\displaystyle{ \mathcal{B} }[/math] if and only if there exists some finer prefilter [math]\displaystyle{ \mathcal{C} \geq \mathcal{B} }[/math] such that [math]\displaystyle{ \mathcal{C} \to x. }[/math]
   • This is the analog of the following false statement: "[math]\displaystyle{ x }[/math] is a cluster point of a sequence if and only if it has a subsequence that converges to [math]\displaystyle{ x }[/math]" (that is, if and only if [math]\displaystyle{ x }[/math] is a subsequential limit).
   • The analog for sequences is false since there is a Hausdorff topology on [math]\displaystyle{ X := [\N \times \N] \cup \{(0, 0)\} }[/math] and a sequence in this space (both defined here) that clusters at [math]\displaystyle{ (0, 0) }[/math] but that also does not have any subsequence that converges to [math]\displaystyle{ (0, 0). }[/math]

Non–equivalence of subnets and subordinate filters

Subnets in the sense of Willard and subnets in the sense of Kelley are the most commonly used definitions of "subnet." The first definition of a subnet ("Kelley–subnet") was introduced by John L. Kelley in 1955. Stephen Willard introduced in 1970 his own variant ("Willard-subnet") of Kelley's definition of subnet. AA–subnets were introduced independently by Smiley (1957), Aarnes and Andenaes (1972), and Murdeshwar (1983); AA–subnets were studied in great detail by Aarnes and Andenaes but they are not often used.

A subset [math]\displaystyle{ R \subseteq I }[/math] of a preordered space [math]\displaystyle{ (I, \leq) }[/math] is frequent or cofinal in [math]\displaystyle{ I }[/math] if for every [math]\displaystyle{ i \in I }[/math] there exists some [math]\displaystyle{ r \in R }[/math] such that [math]\displaystyle{ i \leq r. }[/math] If [math]\displaystyle{ R \subseteq I }[/math] contains a tail of [math]\displaystyle{ I }[/math] then [math]\displaystyle{ R }[/math] is said to be eventual in [math]\displaystyle{ I }[/math]}}; explicitly, this means that there exists some [math]\displaystyle{ i \in I }[/math] such that [math]\displaystyle{ I_{\geq i} \subseteq R }[/math] (that is, [math]\displaystyle{ j \in R }[/math] for all [math]\displaystyle{ j \in I }[/math] satisfying [math]\displaystyle{ i \leq j }[/math]). A subset is eventual if and only if its complement is not frequent (which is termed infrequent). A map [math]\displaystyle{ h : A \to I }[/math] between two preordered sets is order–preserving if whenever [math]\displaystyle{ a, b \in A }[/math] satisfy [math]\displaystyle{ a \leq b, }[/math] then [math]\displaystyle{ h(a) \leq h(b). }[/math]

Definitions: Let [math]\displaystyle{ S = S_\bull ~:~ (A, \leq) \to X \text{ and } N = N_\bull ~:~ (I, \leq) \to X }[/math] be nets. Then

1. [math]\displaystyle{ S_\bull }[/math] is a Willard–subnet of [math]\displaystyle{ N_\bull }[/math] or a subnet in the sense of Willard if there exists an order–preserving map [math]\displaystyle{ h : A \to I }[/math] such that [math]\displaystyle{ S = N \circ h \text{ and } h(A) }[/math] is cofinal in [math]\displaystyle{ I. }[/math]
2. [math]\displaystyle{ S_\bull }[/math] is a Kelley–subnet of [math]\displaystyle{ N_\bull }[/math] or a subnet in the sense of Kelley if there exists a map [math]\displaystyle{ h ~:~ A \to I }[/math] such that [math]\displaystyle{ S = N \circ h }[/math] and whenever [math]\displaystyle{ E \subseteq I }[/math] is eventual in [math]\displaystyle{ I }[/math] then [math]\displaystyle{ h^{-1}(E) }[/math] is eventual in [math]\displaystyle{ A. }[/math]
3. [math]\displaystyle{ S_\bull }[/math] is an AA–subnet of [math]\displaystyle{ N_\bull }[/math] or a subnet in the sense of Aarnes and Andenaes if any of the following equivalent conditions are satisfied:
   1. [math]\displaystyle{ \operatorname{Tails}\left(N_\bull\right) \leq \operatorname{Tails}\left(S_\bull\right). }[/math]
   2. [math]\displaystyle{ \operatorname{TailsFilter}\left(N_\bull\right) \subseteq \operatorname{TailsFilter}\left(S_\bull\right). }[/math]
   3. If [math]\displaystyle{ J }[/math] is eventual in [math]\displaystyle{ I \text{ then } S^{-1}(N(J)) }[/math] is eventual in [math]\displaystyle{ A. }[/math]
   4. For any subset [math]\displaystyle{ R \subseteq X, \text{ if } \operatorname{Tails}\left(S_\bull\right) \text{ and } \{R\} }[/math] mesh, then so do [math]\displaystyle{ \operatorname{Tails}\left(N_\bull\right) \text{ and } \{R\}. }[/math]
   5. For any subset [math]\displaystyle{ R \subseteq X, \text{ if } \operatorname{Tails}\left(S_\bull\right) \leq \{R\} \text{ then } \operatorname{Tails}\left(N_\bull\right) \leq \{R\}. }[/math]

Kelley did not require the map [math]\displaystyle{ h }[/math] to be order preserving while the definition of an AA–subnet does away entirely with any map between the two nets' domains and instead focuses entirely on [math]\displaystyle{ X }[/math] − the nets' common codomain. Every Willard–subnet is a Kelley–subnet and both are AA–subnets. In particular, if [math]\displaystyle{ y_\bull = \left(y_a\right)_{a \in A} }[/math] is a Willard–subnet or a Kelley–subnet of [math]\displaystyle{ x_\bull = \left(x_i\right)_{i \in I} }[/math] then [math]\displaystyle{ \operatorname{Tails}\left(x_\bull\right) \leq \operatorname{Tails}\left(y_\bull\right). }[/math]

Example: If [math]\displaystyle{ I = \N }[/math] and [math]\displaystyle{ x_\bull = \left(x_i\right)_{i \in I} }[/math] is a constant sequence and if [math]\displaystyle{ A = \{1\} }[/math] and [math]\displaystyle{ s_1 := x_1 }[/math] then [math]\displaystyle{ \left(s_a\right)_{a \in A} }[/math] is an AA-subnet of [math]\displaystyle{ x_\bull }[/math] but it is neither a Willard-subnet nor a Kelley-subnet of [math]\displaystyle{ x_\bull. }[/math]

AA–subnets have a defining characterization that immediately shows that they are fully interchangeable with sub(ordinate)filters. Explicitly, what is meant is that the following statement is true for AA–subnets:

If [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{F} }[/math] are prefilters then [math]\displaystyle{ \mathcal{B} \leq \mathcal{F} }[/math] if and only if [math]\displaystyle{ \operatorname{Net}_{\mathcal{F}} }[/math] is an AA–subnet of [math]\displaystyle{ \operatorname{Net}_{\mathcal{B}}. }[/math]

If "AA–subnet" is replaced by "Willard–subnet" or "Kelley–subnet" then the above statement becomes false. In particular, as this counter-example demonstrates, the problem is that the following statement is in general false:

False statement: If [math]\displaystyle{ \mathcal{B} \text{ and } \mathcal{F} }[/math] are prefilters such that [math]\displaystyle{ \mathcal{B} \leq \mathcal{F} \text{ then } \operatorname{Net}_{\mathcal{F}} }[/math] is a Kelley–subnet of [math]\displaystyle{ \operatorname{Net}_{\mathcal{B}}. }[/math]

Since every Willard–subnet is a Kelley–subnet, this statement thus remains false if the word "Kelley–subnet" is replaced with "Willard–subnet".

If "subnet" is defined to mean Willard–subnet or Kelley–subnet then nets and filters are not completely interchangeable because there exists a filter–sub(ordinate)filter relationships that cannot be expressed in terms of a net–subnet relationship between the two induced nets. In particular, the problem is that Kelley–subnets and Willard–subnets are not fully interchangeable with subordinate filters. If the notion of "subnet" is not used or if "subnet" is defined to mean AA–subnet, then this ceases to be a problem and so it becomes correct to say that nets and filters are interchangeable. Despite the fact that AA–subnets do not have the problem that Willard and Kelley subnets have, they are not widely used or known about.

Topologies and prefilters

Throughout, [math]\displaystyle{ (X, \tau) }[/math] is a topological space.

Examples of relationships between filters and topologies

Bases and prefilters

Let [math]\displaystyle{ \mathcal{B} \neq \varnothing }[/math] be a family of sets that covers [math]\displaystyle{ X }[/math] and define [math]\displaystyle{ \mathcal{B}_x = \{B \in \mathcal{B} ~:~ x \in B\} }[/math] for every [math]\displaystyle{ x \in X. }[/math] The definition of a base for some topology can be immediately reworded as: [math]\displaystyle{ \mathcal{B} }[/math] is a base for some topology on [math]\displaystyle{ X }[/math] if and only if [math]\displaystyle{ \mathcal{B}_x }[/math] is a filter base for every [math]\displaystyle{ x \in X. }[/math] If [math]\displaystyle{ \tau }[/math] is a topology on [math]\displaystyle{ X }[/math] and [math]\displaystyle{ \mathcal{B} \subseteq \tau }[/math] then the definitions of [math]\displaystyle{ \mathcal{B} }[/math] is a basis (resp. subbase) for [math]\displaystyle{ \tau }[/math] can be reworded as:

[math]\displaystyle{ \mathcal{B} }[/math] is a base (resp. subbase) for [math]\displaystyle{ \tau }[/math] if and only if for every [math]\displaystyle{ x \in X, \mathcal{B}_x }[/math] is a filter base (resp. filter subbase) that generates the neighborhood filter of [math]\displaystyle{ (X, \tau) }[/math] at [math]\displaystyle{ x. }[/math]

Neighborhood filters

The archetypical example of a filter is the set of all neighborhoods of a point in a topological space. Any neighborhood basis of a point in (or of a subset of) a topological space is a prefilter. In fact, the definition of a neighborhood base can be equivalently restated as: "a neighborhood base is any prefilter that is equivalent the neighborhood filter."

Neighborhood bases at points are examples of prefilters that are fixed but may or may not be principal. If [math]\displaystyle{ X = \R }[/math] has its usual topology and if [math]\displaystyle{ x \in X, }[/math] then any neighborhood filter base [math]\displaystyle{ \mathcal{B} }[/math] of [math]\displaystyle{ x }[/math] is fixed by [math]\displaystyle{ x }[/math] (in fact, it is even true that [math]\displaystyle{ \ker \mathcal{B} = \{x\} }[/math]) but [math]\displaystyle{ \mathcal{B} }[/math] is not principal since [math]\displaystyle{ \{x\} \not\in \mathcal{B}. }[/math] In contrast, a topological space has the discrete topology if and only if the neighborhood filter of every point is a principal filter generated by exactly one point. This shows that a non–principal filter on an infinite set is not necessarily free.

The neighborhood filter of every point [math]\displaystyle{ x }[/math] in topological space [math]\displaystyle{ X }[/math] is fixed since its kernel contains [math]\displaystyle{ x }[/math] (and possibly other points if, for instance, [math]\displaystyle{ X }[/math] is not a T₁ space). This is also true of any neighborhood basis at [math]\displaystyle{ x. }[/math] For any point [math]\displaystyle{ x }[/math] in a T₁ space (for example, a Hausdorff space), the kernel of the neighborhood filter of [math]\displaystyle{ x }[/math] is equal to the singleton set [math]\displaystyle{ \{x\}. }[/math]

However, it is possible for a neighborhood filter at a point to be principal but not discrete (that is, not principal at a single point). A neighborhood basis [math]\displaystyle{ \mathcal{B} }[/math] of a point [math]\displaystyle{ x }[/math] in a topological space is principal if and only if the kernel of [math]\displaystyle{ \mathcal{B} }[/math] is an open set. If in addition the space is T₁ then [math]\displaystyle{ \ker \mathcal{B} = \{x\} }[/math] so that this basis [math]\displaystyle{ \mathcal{B} }[/math] is principal if and only if [math]\displaystyle{ \{x\} }[/math] is an open set.

Generating topologies from filters and prefilters

Suppose [math]\displaystyle{ \mathcal{B} \subseteq \wp(X) }[/math] is not empty (and [math]\displaystyle{ X \neq \varnothing }[/math]). If [math]\displaystyle{ \mathcal{B} }[/math] is a filter on [math]\displaystyle{ X }[/math] then [math]\displaystyle{ \{\varnothing\} \cup \mathcal{B} }[/math] is a topology on [math]\displaystyle{ X }[/math] but the converse is in general false. This shows that in a sense, filters are almost topologies. Topologies of the form [math]\displaystyle{ \{\varnothing\} \cup \mathcal{B} }[/math] where [math]\displaystyle{ \mathcal{B} }[/math] is an ultrafilter on [math]\displaystyle{ X }[/math] are an even more specialized subclass of such topologies; they have the property that every proper subset [math]\displaystyle{ \varnothing \neq S \subseteq X }[/math] is either open or closed, but (unlike the discrete topology) never both. These spaces are, in particular, examples of door spaces.

If [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter (resp. filter subbase, π–system, proper) on [math]\displaystyle{ X }[/math] then the same is true of both [math]\displaystyle{ \{X\} \cup \mathcal{B} }[/math] and the set [math]\displaystyle{ \mathcal{B}_{\cup} }[/math] of all possible unions of one or more elements of [math]\displaystyle{ \mathcal{B}. }[/math] If [math]\displaystyle{ \mathcal{B} }[/math] is closed under finite intersections then the set [math]\displaystyle{ \tau_{\mathcal{B}} = \{\varnothing, X\} \cup \mathcal{B}_{\cup} }[/math] is a topology on [math]\displaystyle{ X }[/math] with both [math]\displaystyle{ \{X\} \cup \mathcal{B}_{\cup} \text{ and } \{X\} \cup \mathcal{B} }[/math] being bases for it. If the π–system [math]\displaystyle{ \mathcal{B} }[/math] covers [math]\displaystyle{ X }[/math] then both [math]\displaystyle{ \mathcal{B}_{\cup} \text{ and } \mathcal{B} }[/math] are also bases for [math]\displaystyle{ \tau_{\mathcal{B}}. }[/math] If [math]\displaystyle{ \tau }[/math] is a topology on [math]\displaystyle{ X }[/math] then [math]\displaystyle{ \tau \setminus \{\varnothing\} }[/math] is a prefilter (or equivalently, a π–system) if and only if it has the finite intersection property (that is, it is a filter subbase), in which case a subset [math]\displaystyle{ \mathcal{B} \subseteq \tau }[/math] will be a basis for [math]\displaystyle{ \tau }[/math] if and only if [math]\displaystyle{ \mathcal{B} \setminus \{\varnothing\} }[/math] is equivalent to [math]\displaystyle{ \tau \setminus \{\varnothing\}, }[/math] in which case [math]\displaystyle{ \mathcal{B} \setminus \{\varnothing\} }[/math] will be a prefilter.

Topologies on directed sets and net convergence

Let [math]\displaystyle{ (I, \leq) }[/math] be a non–empty directed set and let [math]\displaystyle{ \operatorname{Tails}(I) = \left\{I_{\geq i} ~:~ i \in I\right\}, }[/math] where [math]\displaystyle{ I_{\geq i} = \{j \in I ~:~ i \leq j\}. }[/math] Then [math]\displaystyle{ \operatorname{Tails}(I) }[/math] is a prefilter that covers [math]\displaystyle{ I }[/math] and if [math]\displaystyle{ I }[/math] is totally ordered then [math]\displaystyle{ \operatorname{Tails}(I) }[/math] is also closed under finite intersections. This particular prefilter [math]\displaystyle{ \operatorname{Tails}(I) }[/math] forms a base for a topology on [math]\displaystyle{ I }[/math] in which all sets of the form [math]\displaystyle{ I_{\gt i} = \{j \in I ~:~ i \lt j\} }[/math] are also open. The same is true of the topology [math]\displaystyle{ \tau_I := \{\varnothing\} \cup \operatorname{FilterTails}(I) \text{ on } I, }[/math] where [math]\displaystyle{ \operatorname{FilterTails}(I) }[/math] is the filter on [math]\displaystyle{ I }[/math] generated by [math]\displaystyle{ \operatorname{Tails}(I). }[/math] With this topology, convergent nets can be viewed as continuous functions in the following way. Let [math]\displaystyle{ (X, \tau) }[/math] be a topological space, let [math]\displaystyle{ x \in X, }[/math] let [math]\displaystyle{ x_\bull = \left(x_i\right)_{i \in I} ~:~ I \to X }[/math] be a net in [math]\displaystyle{ X, }[/math] and let [math]\displaystyle{ \tau(x) \subseteq \tau }[/math] denote the set of all open neighborhoods of [math]\displaystyle{ x. }[/math] If the net [math]\displaystyle{ x_\bull }[/math] converges to [math]\displaystyle{ x \text{ in } (X, \tau) }[/math] then [math]\displaystyle{ x_\bull ~:~ \left(I, \tau_I\right) \to \left(X, \{\varnothing\} \cup \tau(x)\right) }[/math] is necessarily continuous although in general, the converse is false (for example, consider if [math]\displaystyle{ x_\bull }[/math] is constant and not equal to [math]\displaystyle{ x }[/math]). But if in addition to continuity, the preimage under [math]\displaystyle{ x_\bull }[/math] of every [math]\displaystyle{ N \in \tau(x) }[/math] is not empty, then the net [math]\displaystyle{ x_\bull }[/math] will necessarily converge to [math]\displaystyle{ x \text{ in } (X, \tau). }[/math] In this way, the empty set is all that separates net convergence and continuity.

Another way in which a convergent nets can be viewed as continuous functions is, for any given [math]\displaystyle{ x \in X }[/math] and net [math]\displaystyle{ x_\bull = \left(x_i\right)_{i \in I} ~:~ I \to X, }[/math] to first extend the net to a new net [math]\displaystyle{ \hat{x}_\bull := \left(\hat{x}_i\right)_{i \in I} : I \cup \{\infty\} \to X, }[/math] where [math]\displaystyle{ \infty \not\in I }[/math] is a new symbol, by defining [math]\displaystyle{ \hat{x}_\infty := x \text{ and } \hat{x}_i := x_i }[/math] for every [math]\displaystyle{ i \in I. }[/math] If [math]\displaystyle{ I \cup \{\infty\} }[/math] is endowed with the topology [math]\displaystyle{ \tau_{I \cup \{\infty\}} ~:=~ \wp(I) ~\cup~ \left\{\,\{\infty\} \cup S ~:~ S \in \operatorname{FilterTails}(I)\right\} ~=~ \wp(I) ~\cup~ \left(\,\{\infty\} \,(\cup)\, \operatorname{FilterTails}(I)\,\right) }[/math] then [math]\displaystyle{ x_\bull \to x \text{ in } X }[/math] (that is, the net [math]\displaystyle{ x_\bull }[/math] converges to [math]\displaystyle{ x }[/math]) if and only if [math]\displaystyle{ \hat{x}_\bull : \left(I \cup \{\infty\}, \tau_{I \cup \{\infty\}}\right) \to (X, \tau) }[/math] is a continuous function. Moreover, [math]\displaystyle{ I }[/math] is always a dense subset of [math]\displaystyle{ I \cup \{\infty\}. }[/math]

Topological properties and prefilters

Neighborhoods and topologies

The neighborhood filter of a nonempty subset [math]\displaystyle{ S \subseteq X }[/math] in a topological space [math]\displaystyle{ X }[/math] is equal to the intersection of all neighborhood filters of all points in [math]\displaystyle{ S. }[/math] A subset [math]\displaystyle{ S \subseteq X }[/math] is open in [math]\displaystyle{ X }[/math] if and only if whenever [math]\displaystyle{ \mathcal{F} }[/math] is a filter on [math]\displaystyle{ X }[/math] and [math]\displaystyle{ s \in S, }[/math] then [math]\displaystyle{ \mathcal{F} \to s \text{ in } X \text{ implies } S \in \mathcal{F}. }[/math]

Suppose [math]\displaystyle{ \sigma \text{ and } \tau }[/math] are topologies on [math]\displaystyle{ X. }[/math] Then [math]\displaystyle{ \tau }[/math] is finer than [math]\displaystyle{ \sigma }[/math] (that is, [math]\displaystyle{ \sigma \subseteq \tau }[/math]) if and only if whenever [math]\displaystyle{ x \in X \text{ and } \mathcal{B} }[/math] is a filter on [math]\displaystyle{ X, }[/math] if [math]\displaystyle{ \mathcal{B} \to x \text{ in } (X, \tau) }[/math] then [math]\displaystyle{ \mathcal{B} \to x \text{ in } (X, \sigma). }[/math] Consequently, [math]\displaystyle{ \sigma = \tau }[/math] if and only if for every filter [math]\displaystyle{ \mathcal{B} \text{ on } X }[/math] and every [math]\displaystyle{ x \in X, \mathcal{B} \to x \text{ in } (X, \sigma) }[/math] if and only if [math]\displaystyle{ \mathcal{B} \to x \text{ in } (X, \tau). }[/math] However, it is possible that [math]\displaystyle{ \sigma \neq \tau }[/math] while also for every filter [math]\displaystyle{ \mathcal{B} \text{ on } X, \mathcal{B} }[/math] converges to some point of [math]\displaystyle{ X \text{ in } (X, \sigma) }[/math] if and only if [math]\displaystyle{ \mathcal{B} }[/math] converges to some point of [math]\displaystyle{ X \text{ in } (X, \tau). }[/math]

Closure

If [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter on a subset [math]\displaystyle{ S \subseteq X }[/math] then every cluster point of [math]\displaystyle{ \mathcal{B} \text{ in } X }[/math] belongs to [math]\displaystyle{ \operatorname{cl}_X S. }[/math]

If [math]\displaystyle{ x \in X \text{ and } S \subseteq X }[/math] is a non-empty subset, then the following are equivalent:

1. [math]\displaystyle{ x \in \operatorname{cl}_X S }[/math]
2. [math]\displaystyle{ x }[/math] is a limit point of a prefilter on [math]\displaystyle{ S. }[/math] Explicitly: there exists a prefilter [math]\displaystyle{ \mathcal{F} \subseteq \wp(S) \text{ on } S }[/math] such that [math]\displaystyle{ \mathcal{F} \to x \text{ in } X. }[/math]
3. [math]\displaystyle{ x }[/math] is a limit point of a filter on [math]\displaystyle{ S. }[/math]
4. There exists a prefilter [math]\displaystyle{ \mathcal{F} \text{ on } X }[/math] such that [math]\displaystyle{ S \in \mathcal{F} \text{ and } \mathcal{F} \to x \text{ in } X. }[/math]
5. The prefilter [math]\displaystyle{ \{S\} }[/math] meshes with the neighborhood filter [math]\displaystyle{ \mathcal{N}(x). }[/math] Said differently, [math]\displaystyle{ x }[/math] is a cluster point of the prefilter [math]\displaystyle{ \{S\}. }[/math]
6. The prefilter [math]\displaystyle{ \{S\} }[/math] meshes with some (or equivalently, with every) filter base for [math]\displaystyle{ \mathcal{N}(x) }[/math] (that is, with every neighborhood basis at [math]\displaystyle{ x }[/math]).

The following are equivalent:

1. [math]\displaystyle{ x }[/math] is a limit points of [math]\displaystyle{ S \text{ in } X. }[/math]
2. There exists a prefilter [math]\displaystyle{ \mathcal{F} \subseteq \wp(S) \text{ on } \{S\} \setminus \{x\} }[/math] such that [math]\displaystyle{ \mathcal{F} \to x \text{ in } X. }[/math]

Closed sets

If [math]\displaystyle{ S \subseteq X }[/math] is not empty then the following are equivalent:

1. [math]\displaystyle{ S }[/math] is a closed subset of [math]\displaystyle{ X. }[/math]
2. If [math]\displaystyle{ x \in X \text{ and } \mathcal{F} \subseteq \wp(S) }[/math] is a prefilter on [math]\displaystyle{ S }[/math] such that [math]\displaystyle{ \mathcal{F} \to x \text{ in } X, }[/math] then [math]\displaystyle{ x \in S. }[/math]
3. If [math]\displaystyle{ x \in X \text{ and } \mathcal{F} \subseteq \wp(S) }[/math] is a prefilter on [math]\displaystyle{ S }[/math] such that [math]\displaystyle{ x }[/math] is an accumulation points of [math]\displaystyle{ \mathcal{F} \text{ in } X, }[/math] then [math]\displaystyle{ x \in S. }[/math]
4. If [math]\displaystyle{ x \in X }[/math] is such that the neighborhood filter [math]\displaystyle{ \mathcal{N}(x) }[/math] meshes with [math]\displaystyle{ \{S\} }[/math] then [math]\displaystyle{ x \in S. }[/math]

Hausdorffness

The following are equivalent:

1. [math]\displaystyle{ X }[/math] is a Hausdorff space.
2. Every prefilter on [math]\displaystyle{ X }[/math] converges to at most one point in [math]\displaystyle{ X. }[/math]
3. The above statement but with the word "prefilter" replaced by any one of the following: filter, ultra prefilter, ultrafilter.

Compactness

As discussed in this article, the Ultrafilter Lemma is closely related to many important theorems involving compactness.

The following are equivalent:

1. [math]\displaystyle{ (X, \tau) }[/math] is a compact space.
2. Every ultrafilter on [math]\displaystyle{ X }[/math] converges to at least one point in [math]\displaystyle{ X. }[/math]
   • That this condition implies compactness can be proven by using only the ultrafilter lemma. That compactness implies this condition can be proven without the ultrafilter lemma (or even the axiom of choice).
3. The above statement but with the word "ultrafilter" replaced by "ultra prefilter".
4. For every filter [math]\displaystyle{ \mathcal{C} \text{ on } X }[/math] there exists a filter [math]\displaystyle{ \mathcal{F} \text{ on } X }[/math] such that [math]\displaystyle{ \mathcal{C} \leq \mathcal{F} }[/math] and [math]\displaystyle{ \mathcal{F} }[/math] converges to some point of [math]\displaystyle{ X. }[/math]
5. The above statement but with each instance of the word "filter" replaced by: prefilter.
6. Every filter on [math]\displaystyle{ X }[/math] has at least one cluster point in [math]\displaystyle{ X. }[/math]
   • That this condition is equivalent to compactness can be proven by using only the ultrafilter lemma.
7. The above statement but with the word "filter" replaced by "prefilter".
8. Alexander subbase theorem: There exists a subbase [math]\displaystyle{ \mathcal{S} \text{ for } \tau }[/math] such that every cover of [math]\displaystyle{ X }[/math] by sets in [math]\displaystyle{ \mathcal{S} }[/math] has a finite subcover.
   • That this condition is equivalent to compactness can be proven by using only the ultrafilter lemma.

If [math]\displaystyle{ \mathcal{F} }[/math] is the set of all complements of compact subsets of a given topological space [math]\displaystyle{ X, }[/math] then [math]\displaystyle{ \mathcal{F} }[/math] is a filter on [math]\displaystyle{ X }[/math] if and only if [math]\displaystyle{ X }[/math] is not compact.

Theorem — If [math]\displaystyle{ \mathcal{B} }[/math] is a filter on a compact space and [math]\displaystyle{ C }[/math] is the set of cluster points of [math]\displaystyle{ \mathcal{B}, }[/math] then every neighborhood of [math]\displaystyle{ C }[/math] belongs to [math]\displaystyle{ \mathcal{B}. }[/math] Thus a filter on a compact Hausdorff space converges if and only if it has a single cluster point.

Continuity

Let [math]\displaystyle{ f : X \to Y }[/math] be a map between topological spaces [math]\displaystyle{ (X, \tau) \text{ and } (Y, \upsilon). }[/math]

Given [math]\displaystyle{ x \in X, }[/math] the following are equivalent:

1. [math]\displaystyle{ f : X \to Y }[/math] is continuous at [math]\displaystyle{ x. }[/math]
2. Definition: For every neighborhood [math]\displaystyle{ V }[/math] of [math]\displaystyle{ f(x) \text{ in } Y }[/math] there exists some neighborhood [math]\displaystyle{ N }[/math] of [math]\displaystyle{ x \text{ in } X }[/math] such that [math]\displaystyle{ f(N) \subseteq V. }[/math]
3. [math]\displaystyle{ f(\mathcal{N}(x)) \to f(x) \text{ in } Y. }[/math]
4. If [math]\displaystyle{ \mathcal{B} }[/math] is a filter on [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ \mathcal{B} \to x \text{ in } X }[/math] then [math]\displaystyle{ f(\mathcal{B}) \to f(x) \text{ in } Y. }[/math]
5. The above statement but with the word "filter" replaced by "prefilter".

The following are equivalent:

1. [math]\displaystyle{ f : X \to Y }[/math] is continuous.
2. If [math]\displaystyle{ x \in X \text{ and } \mathcal{B} }[/math] is a prefilter on [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ \mathcal{B} \to x \text{ in } X }[/math] then [math]\displaystyle{ f(\mathcal{B}) \to f(x) \text{ in } Y. }[/math]
3. If [math]\displaystyle{ x \in X }[/math] is a limit point of a prefilter [math]\displaystyle{ \mathcal{B} \text{ on } X }[/math] then [math]\displaystyle{ f(x) }[/math] is a limit point of [math]\displaystyle{ f(\mathcal{B}) \text{ in } Y. }[/math]
4. Any one of the above two statements but with the word "prefilter" replaced by "filter".

If [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter on [math]\displaystyle{ X, x \in X }[/math] is a cluster point of [math]\displaystyle{ \mathcal{B}, \text{ and } f : X \to Y }[/math] is continuous, then [math]\displaystyle{ f(x) }[/math] is a cluster point in [math]\displaystyle{ Y }[/math] of the prefilter [math]\displaystyle{ f(\mathcal{B}). }[/math]

A subset [math]\displaystyle{ D }[/math] of a topological space [math]\displaystyle{ X }[/math] is dense in [math]\displaystyle{ X }[/math] if and only if for every [math]\displaystyle{ x \in X, }[/math] the trace [math]\displaystyle{ \mathcal{N}_X(x)\big\vert_D }[/math] of the neighborhood filter [math]\displaystyle{ \mathcal{N}_X(x) }[/math] along [math]\displaystyle{ D }[/math] does not contain the empty set (in which case it will be a filter on [math]\displaystyle{ D }[/math]).

Suppose [math]\displaystyle{ f : D \to Y }[/math] is a continuous map into a Hausdorff regular space [math]\displaystyle{ Y }[/math] and that [math]\displaystyle{ D }[/math] is a dense subset of a topological space [math]\displaystyle{ X. }[/math] Then [math]\displaystyle{ f }[/math] has a continuous extension [math]\displaystyle{ F : X \to Y }[/math] if and only if for every [math]\displaystyle{ x \in X, }[/math] the prefilter [math]\displaystyle{ f\left(\mathcal{N}_X(x)\big\vert_D\right) }[/math] converges to some point in [math]\displaystyle{ Y. }[/math] Furthermore, this continuous extension will be unique whenever it exists.

Products

Suppose [math]\displaystyle{ X_\bull := \left(X_i\right)_{i \in I} }[/math] is a non–empty family of non–empty topological spaces and that is a family of prefilters where each [math]\displaystyle{ \mathcal{B}_i }[/math] is a prefilter on [math]\displaystyle{ X_i. }[/math] Then the product [math]\displaystyle{ \mathcal{B}_\bull }[/math] of these prefilters (defined above) is a prefilter on the product space [math]\displaystyle{ {\textstyle\prod} X_\bull, }[/math] which as usual, is endowed with the product topology.

If [math]\displaystyle{ x_\bull := \left(x_i\right)_{i \in I} \in {\textstyle\prod} X_\bull, }[/math] then [math]\displaystyle{ \mathcal{B}_\bull \to x_\bull \text{ in } {\textstyle\prod} X_\bull }[/math] if and only if [math]\displaystyle{ \mathcal{B}_i \to x_i \text{ in } X_i \text{ for every } i \in I. }[/math]

Suppose [math]\displaystyle{ X \text{ and } Y }[/math] are topological spaces, [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter on [math]\displaystyle{ X }[/math] having [math]\displaystyle{ x \in X }[/math] as a cluster point, and [math]\displaystyle{ \mathcal{C} }[/math] is a prefilter on [math]\displaystyle{ Y }[/math] having [math]\displaystyle{ y \in Y }[/math] as a cluster point. Then [math]\displaystyle{ (x, y) }[/math] is a cluster point of [math]\displaystyle{ \mathcal{B} \times \mathcal{C} }[/math] in the product space [math]\displaystyle{ X \times Y. }[/math] However, if [math]\displaystyle{ X = Y = \Q }[/math] then there exist sequences [math]\displaystyle{ \left(x_i\right)_{i=1}^\infty \subseteq X \text{ and } \left(y_i\right)_{i=1}^\infty \subseteq Y }[/math] such that both of these sequences have a cluster point in [math]\displaystyle{ \Q }[/math] but the sequence [math]\displaystyle{ \left(x_i, y_i\right)_{i=1}^\infty \subseteq X \times Y }[/math] does not have a cluster point in [math]\displaystyle{ X \times Y. }[/math]

Example application: The ultrafilter lemma along with the axioms of ZF imply Tychonoff's theorem for compact Hausdorff spaces:

Proof

Let [math]\displaystyle{ X_\bull := \left(X_i\right)_{i \in I} }[/math] be compact Hausdorff topological spaces. Assume that the ultrafilter lemma holds (because of the Hausdorff assumption, this proof does not need the full strength of the axiom of choice; the ultrafilter lemma suffices). Let [math]\displaystyle{ X := {\textstyle\prod} X_\bull }[/math] be given the product topology (which makes [math]\displaystyle{ X }[/math] a Hausdorff space) and for every [math]\displaystyle{ i, }[/math] let [math]\displaystyle{ \Pr{}_i : X \to X_i }[/math] denote this product's projections. If [math]\displaystyle{ X = \varnothing }[/math] then [math]\displaystyle{ X }[/math] is compact and the proof is complete so assume [math]\displaystyle{ X \neq \varnothing. }[/math] Despite the fact that [math]\displaystyle{ X \neq \varnothing, }[/math] because the axiom of choice is not assumed, the projection maps [math]\displaystyle{ \Pr{}_i : X \to X_i }[/math] are not guaranteed to be surjective. Let [math]\displaystyle{ \mathcal{B} }[/math] be an ultrafilter on [math]\displaystyle{ X }[/math] and for every [math]\displaystyle{ i, }[/math] let [math]\displaystyle{ \mathcal{B}_i }[/math] denote the ultrafilter on [math]\displaystyle{ X_i }[/math] generated by the ultra prefilter [math]\displaystyle{ \Pr{}_i(\mathcal{B}). }[/math] Because [math]\displaystyle{ X_i }[/math] is compact and Hausdorff, the ultrafilter [math]\displaystyle{ \mathcal{B}_i }[/math] converges to a unique limit point [math]\displaystyle{ x_i \in X_i }[/math] (because of [math]\displaystyle{ x_i }[/math]'s uniqueness, this definition does not require the axiom of choice). Let [math]\displaystyle{ x := \left(x_i\right)_{i \in I} }[/math] where [math]\displaystyle{ x }[/math] satisfies [math]\displaystyle{ \Pr{}_i(x) = x_i }[/math] for every [math]\displaystyle{ i. }[/math] The characterization of convergence in the product topology that was given above implies that [math]\displaystyle{ \mathcal{B} \to x \text{ in } X. }[/math] Thus every ultrafilter on [math]\displaystyle{ X }[/math] converges to some point of [math]\displaystyle{ X, }[/math] which implies that [math]\displaystyle{ X }[/math] is compact (recall that this implication's proof only required the ultrafilter lemma). [math]\displaystyle{ \blacksquare }[/math]

Examples of applications of prefilters

Uniformities and Cauchy prefilters

A uniform space is a set [math]\displaystyle{ X }[/math] equipped with a filter on [math]\displaystyle{ X \times X }[/math] that has certain properties. A base or fundamental system of entourages is a prefilter on [math]\displaystyle{ X \times X }[/math] whose upward closure is a uniform space. A prefilter [math]\displaystyle{ \mathcal{B} }[/math] on a uniform space [math]\displaystyle{ X }[/math] with uniformity [math]\displaystyle{ \mathcal{F} }[/math] is called a Cauchy prefilter if for every entourage [math]\displaystyle{ N \in \mathcal{F}, }[/math] there exists some [math]\displaystyle{ B \in \mathcal{B} }[/math] that is [math]\displaystyle{ N }[/math]–small, which means that [math]\displaystyle{ B \times B \subseteq N. }[/math] A minimal Cauchy filter is a minimal element (with respect to [math]\displaystyle{ \,\leq\, }[/math] or equivalently, to [math]\displaystyle{ \,\subseteq }[/math]) of the set of all Cauchy filters on [math]\displaystyle{ X. }[/math] Examples of minimal Cauchy filters include the neighborhood filter [math]\displaystyle{ \mathcal{N}_X(x) }[/math] of any point [math]\displaystyle{ x \in X. }[/math] Every convergent filter on a uniform space is Cauchy. Moreover, every cluster point of a Cauchy filter is a limit point.

A uniform space [math]\displaystyle{ (X, \mathcal{F}) }[/math] is called complete (resp. sequentially complete) if every Cauchy prefilter (resp. every elementary Cauchy prefilter) on [math]\displaystyle{ X }[/math] converges to at least one point of [math]\displaystyle{ X }[/math] (replacing all instance of the word "prefilter" with "filter" results in equivalent statement). Every compact uniform space is complete because any Cauchy filter has a cluster point (by compactness), which is necessarily also a limit point (since the filter is Cauchy).

Uniform spaces were the result of attempts to generalize notions such as "uniform continuity" and "uniform convergence" that are present in metric spaces. Every topological vector space, and more generally, every topological group can be made into a uniform space in a canonical way. Every uniformity also generates a canonical induced topology. Filters and prefilters play an important role in the theory of uniform spaces. For example, the completion of a Hausdorff uniform space (even if it is not metrizable) is typically constructed by using minimal Cauchy filters. Nets are less ideal for this construction because their domains are extremely varied (for example, the class of all Cauchy nets is not a set); sequences cannot be used in the general case because the topology might not be metrizable, first–countable, or even sequential. The set of all minimal Cauchy filters on a Hausdorff topological vector space (TVS) [math]\displaystyle{ X }[/math] can made into a vector space and topologized in such a way that it becomes a completion of [math]\displaystyle{ X }[/math] (with the assignment [math]\displaystyle{ x \mapsto \mathcal{N}_X(x) }[/math] becoming a linear topological embedding that identifies [math]\displaystyle{ X }[/math] as a dense vector subspace of this completion).

More generally, a Cauchy space is a pair [math]\displaystyle{ (X, \mathfrak{C}) }[/math] consisting of a set [math]\displaystyle{ X }[/math] together a family [math]\displaystyle{ \mathfrak{C} \subseteq \wp(\wp(X)) }[/math] of (proper) filters, whose members are declared to be "Cauchy filters", having all of the following properties:

1. For each [math]\displaystyle{ x \in X, }[/math] the discrete ultrafilter at [math]\displaystyle{ x }[/math] is an element of [math]\displaystyle{ \mathfrak{C}. }[/math]
2. If [math]\displaystyle{ F \in \mathfrak{C} }[/math] is a subset of a proper filter [math]\displaystyle{ G, }[/math] then [math]\displaystyle{ G \in \mathfrak{C}. }[/math]
3. If [math]\displaystyle{ F, G \in \mathfrak{C} }[/math] and if each member of [math]\displaystyle{ F }[/math] intersects each member of [math]\displaystyle{ G, }[/math] then [math]\displaystyle{ F \cap G \in \mathfrak{C}. }[/math]

The set of all Cauchy filters on a uniform space forms a Cauchy space. Every Cauchy space is also a convergence space. A map [math]\displaystyle{ f : X \to Y }[/math] between two Cauchy spaces is called Cauchy continuous if the image of every Cauchy filter in [math]\displaystyle{ X }[/math] is a Cauchy filter in [math]\displaystyle{ Y. }[/math] Unlike the category of topological spaces, the category of Cauchy spaces and Cauchy continuous maps is Cartesian closed, and contains the category of proximity spaces.

Convergence of nets of sets

There is often a personal preference of nets over filters or filters over nets. This example shows that the choice between nets and filters is not a dichotomy by combining them together.

A net of sets in [math]\displaystyle{ X }[/math] or a net of subsets of [math]\displaystyle{ X }[/math] refers to a net in the power set [math]\displaystyle{ \wp(X) }[/math] of [math]\displaystyle{ X; }[/math] that is, a net of sets in [math]\displaystyle{ X }[/math] is a function from a non–empty directed set into [math]\displaystyle{ \wp(X). }[/math] However, a "net in [math]\displaystyle{ X }[/math]" will always refer to a net valued in [math]\displaystyle{ X }[/math] and never to a net valued in [math]\displaystyle{ \wp(X) }[/math] although for emphasis or contrast, a net in [math]\displaystyle{ X }[/math] may also be referred to as a net of points in [math]\displaystyle{ X }[/math]. A net [math]\displaystyle{ S_\bull = \left(S_i\right)_{i \in I} }[/math] of sets in [math]\displaystyle{ X }[/math] is called a net of singleton (resp. non–empty, finite, compact, etc.) sets in [math]\displaystyle{ X }[/math] if every [math]\displaystyle{ S_i }[/math] has this property. Similarly, [math]\displaystyle{ S_\bull }[/math] is called eventually empty (resp. non–empty, finite, compact, etc.) if there is some index [math]\displaystyle{ i }[/math] such that this is true of [math]\displaystyle{ S_j }[/math] for every index [math]\displaystyle{ j \geq i. }[/math]

By definition, a net [math]\displaystyle{ S_\bull = \left(S_i\right)_{i \in I} }[/math] of sets converges to (respectively, clusters at) a point or subset if and only if the same is true of its set/family of tails [math]\displaystyle{ \operatorname{Tails}\left(S_\bull\right) := \left\{S_{\geq i} : i \in I\right\} }[/math] where for every index [math]\displaystyle{ i }[/math] the set [math]\displaystyle{ S_{\geq i} := \bigcup_{i \, \leq \, j \, \in \, I} S_j. }[/math] is called the tail of [math]\displaystyle{ S_\bull }[/math] starting at [math]\displaystyle{ i }[/math] (a definition generalizes that of a tail of a net of points).

The following examples show some of the many ways that nets of sets arise naturally.

Example: Prefilters as nets of sets

If [math]\displaystyle{ \mathcal{B} }[/math] is a prefilter on [math]\displaystyle{ X }[/math] then [math]\displaystyle{ (\mathcal{B}, \supseteq) }[/math] is a (partially ordered) directed set, so that the identity map [math]\displaystyle{ \operatorname{id}_{\mathcal{B}} : (\mathcal{B}, \supseteq) \to \mathcal{B} }[/math] is a net of sets in [math]\displaystyle{ X. }[/math] Every prefilter can be canonically identified with this net of sets (that is, with its identity map when the prefilter/domain is directed by [math]\displaystyle{ \supseteq }[/math]). Thus it is significantly easier to canonically associate every prefilter with a net of sets than with a net of points (as was done above), and because the relationship is also much simpler, it is easier utilize. For instance, it is readily seen that the tail of the net [math]\displaystyle{ \operatorname{id}_{\mathcal{B}} }[/math] starting at a given index [math]\displaystyle{ B \in \mathcal{B} }[/math] is equal to [math]\displaystyle{ B }[/math] (in other words, the tail starting at an index is the index itself) so that [math]\displaystyle{ \operatorname{Tails}\left(\operatorname{id}_{\mathcal{B}}\right) = \mathcal{B} }[/math] (that is, this net's tails are its indices) and so the prefilter [math]\displaystyle{ \mathcal{B} }[/math] converges to (respectively, clusters at) a given point or subset if and only if the same is true of its canonical net of sets [math]\displaystyle{ \operatorname{id}_{\mathcal{B}}. }[/math] In particular, information (including intuition and visualizations) about how or why a prefilter [math]\displaystyle{ \mathcal{B} }[/math] converges to (or doesn't converge to, or clusters at, etc.) a point or set can almost immediately be obtained from information about how/why the net of sets [math]\displaystyle{ \operatorname{id}_{\mathcal{B}} }[/math] does the same (or vice versa).

Example: Nets of points as nets of sets

Consideration of the following bijective correspondence leads naturally to the above definitions of convergence and clustering for a net of sets, which are defined analogously to the original definitions given for a net of points.

(Nets of points [math]\displaystyle{ \leftrightarrow }[/math] Nets of singleton sets): Every net [math]\displaystyle{ x_\bull = \left(x_i\right)_{i \in I} }[/math] of points can be uniquely associated with the canonical net of singleton sets [math]\displaystyle{ \left(\left\{x_i\right\}\right)_{i \in I} }[/math] and conversely, every net of singleton sets is uniquely associated with a canonical net of points (defined in the obvious way). The tail of [math]\displaystyle{ \left(\left\{x_i\right\}\right)_{i \in I} }[/math] starting at an index [math]\displaystyle{ i }[/math] is equal to that of [math]\displaystyle{ x_\bull }[/math] (that is, equal to [math]\displaystyle{ x_{\geq i} }[/math]); consequently, [math]\displaystyle{ \operatorname{Tails}\left(x_\bull\right) = \operatorname{Tails}\left(\left(\left\{x_i\right\}\right)_{i \in I}\right). }[/math] This makes it apparent that the definition of "convergence of a net of sets" in [math]\displaystyle{ X }[/math] is indeed a generalization of the original definition of "convergence of a net of points" in [math]\displaystyle{ X }[/math] (because [math]\displaystyle{ x_\bull \to R }[/math] if and only if [math]\displaystyle{ \left(\left\{x_i\right\}\right)_{i \in I} \to R }[/math]). The same is true of the definition of "clustering of a net of sets" because a net of points clusters at a given point or subset (according to the original definition) if and only if this is true of its associated net of singleton sets.

Example: Tails of nets as nets of sets

If [math]\displaystyle{ x_\bull = \left(x_i\right)_{i \in I} }[/math] is a net of points or sets then the assignment [math]\displaystyle{ i \mapsto x_{\geq i} }[/math] that sends an index to the tail starting at that index, gives rise to the net of tails [math]\displaystyle{ x_{\geq \bull} = \left(x_{\geq i}\right)_{i \in I}. }[/math] The tail of [math]\displaystyle{ x_{\geq \bull} }[/math] starting at a given index [math]\displaystyle{ i \in I }[/math] is equal to that of [math]\displaystyle{ x_\bull }[/math] (both equal [math]\displaystyle{ x_{\geq i}; }[/math] and consequently, [math]\displaystyle{ x_{\geq \bull} }[/math] is its own net of tails). Because [math]\displaystyle{ \operatorname{Tails}\left(x_\bull\right) = \operatorname{Tails}\left(x_{\geq \bull}\right), }[/math] the net [math]\displaystyle{ x_\bull }[/math] converges to (respectively, clusters at) some given point or subset if and only if the same is true of its net of tails [math]\displaystyle{ x_{\geq \bull}. }[/math]

Example: Pull-backs and push-forwards of nets as nets of sets

Nets of sets arise naturally when pulling back nets in a function's codomain, and when pushing forward a net of sets in its domain. If [math]\displaystyle{ f : X \to Y }[/math] is a map and [math]\displaystyle{ y_\bull = \left(y_i\right)_{i \in I} }[/math] is a net of sets or points then let [math]\displaystyle{ f^{-1}\left(y_\bull\right) := \left(f^{-1}\left(y_i\right)\right)_{i \in I} \quad \text{ and } \quad f\left(y_\bull\right) := \left(f\left(y_i\right)\right)_{i \in I} }[/math] so that [math]\displaystyle{ f^{-1}\left(y_\bull\right) }[/math] denotes the net of sets [math]\displaystyle{ I \to \wp(X) }[/math] defined by [math]\displaystyle{ i \mapsto f^{-1}\left(y_i\right). }[/math] In particular, [math]\displaystyle{ y_\bull }[/math] eventually being contained in [math]\displaystyle{ f(X) }[/math] (meaning that [math]\displaystyle{ y_{\geq i} \subseteq f(X) }[/math] for some [math]\displaystyle{ i }[/math]) is not a necessary condition for [math]\displaystyle{ f^{-1}\left(y_\bull\right) }[/math] to be a net of set. So even if a net [math]\displaystyle{ \left(y_i\right)_{i \in I} }[/math] of points in [math]\displaystyle{ Y }[/math] cannot be pulled back by [math]\displaystyle{ f }[/math] to a net [math]\displaystyle{ \left(x_i\right)_{i \in I} }[/math] of points in [math]\displaystyle{ X }[/math] (say because it is not entirely/eventually in the image of [math]\displaystyle{ f }[/math]), it is nevertheless still possible to talk about the net of sets [math]\displaystyle{ f^{-1}\left(y_\bull\right) }[/math] and its properties (such as convergence or clustering).

Properties of tails of nets of sets

Suppose [math]\displaystyle{ S_\bull = \left(S_i\right)_{i \in I} }[/math] is a net of sets in [math]\displaystyle{ X. }[/math] The family [math]\displaystyle{ \operatorname{Tails}\left(S_\bull\right) }[/math] is a prefilter if and only if it does not contain the empty set, which is equivalent to [math]\displaystyle{ S_\bull }[/math] not being eventually empty; in this case the upward closure in [math]\displaystyle{ X }[/math] of this prefilter of tails is called the filter of tails or eventuality filter in [math]\displaystyle{ X }[/math] generated by [math]\displaystyle{ S_\bull. }[/math] A net [math]\displaystyle{ y_\bull }[/math] (of sets or points) is eventually contained in a set [math]\displaystyle{ C }[/math] if and only if [math]\displaystyle{ \{C\} \leq \operatorname{Tails}\left(y_\bull\right); }[/math] so [math]\displaystyle{ S_\bull }[/math] is eventually empty if and only if [math]\displaystyle{ \{\varnothing\} \leq \operatorname{Tails}\left(S_\bull\right). }[/math]

Suppose [math]\displaystyle{ f : X \to Y }[/math] is a map and [math]\displaystyle{ y_\bull = \left(y_i\right)_{i \in I} }[/math] is a net of sets (or points). The tail of [math]\displaystyle{ f\left(y_\bull\right) }[/math] starting at an index [math]\displaystyle{ i }[/math] is equal to [math]\displaystyle{ f\left(y_{\geq i}\right) }[/math] and similarly, the tail of [math]\displaystyle{ f^{-1}\left(y_\bull\right) }[/math] starting at [math]\displaystyle{ i }[/math] is [math]\displaystyle{ f^{-1}\left(y_{\geq i}\right). }[/math] Consequently, [math]\displaystyle{ \operatorname{Tails}\left(f\left(y_\bull\right)\right) = f\left(\operatorname{Tails}\left(y_\bull\right)\right) }[/math] where this family is a prefilter if and only if [math]\displaystyle{ \operatorname{Tails}\left(y_\bull\right) }[/math] is a prefilter; similarly, [math]\displaystyle{ \operatorname{Tails}\left(f^{-1}\left(y_\bull\right)\right) = f^{-1}\left(\operatorname{Tails}\left(y_\bull\right)\right). }[/math] One useful consequence of this definition is that [math]\displaystyle{ \operatorname{Tails}\left(f^{-1}\left(y_\bull\right)\right) }[/math] is a prefilter if and only if [math]\displaystyle{ y_\bull }[/math] cofinally intersects (or for points, is cofinally in) [math]\displaystyle{ f(X), }[/math] meaning that for every index [math]\displaystyle{ i, }[/math] there is some [math]\displaystyle{ j \geq i }[/math] such that [math]\displaystyle{ y_j \cap f(X) \neq \varnothing }[/math] (where this intersection means [math]\displaystyle{ y_j \in f(X) }[/math] if [math]\displaystyle{ y_j }[/math] is a point instead of a set).

Convergence and clustering

A net of sets [math]\displaystyle{ S_\bull }[/math] is said to converge in [math]\displaystyle{ (X, \tau) }[/math] to a given point or subset [math]\displaystyle{ x }[/math] of [math]\displaystyle{ X, }[/math] written [math]\displaystyle{ S_\bull \to x \text{ in } (X, \tau), }[/math] if [math]\displaystyle{ \,\operatorname{Tails}\left(S_\bull\right) \to x \text{ in } (X, \tau), }[/math] which recall was defined to mean that [math]\displaystyle{ \mathcal{N}_\tau(x) \leq \operatorname{Tails}\left(S_\bull\right). }[/math] Explicitly, this happens if and only if for every neighborhood [math]\displaystyle{ U }[/math] of [math]\displaystyle{ x, }[/math] there exists some index [math]\displaystyle{ i }[/math] such that [math]\displaystyle{ S_{\geq i} \subseteq U. }[/math] Similarly, [math]\displaystyle{ S_\bull }[/math] is said to cluster at a given point or subset [math]\displaystyle{ x }[/math] of [math]\displaystyle{ X }[/math] if [math]\displaystyle{ \operatorname{Tails}\left(S_\bull\right) }[/math] meshes with [math]\displaystyle{ \mathcal{N}_\tau(x) }[/math] (written [math]\displaystyle{ \mathcal{N}_\tau(x) \;\#\; \operatorname{Tails}\left(S_\bull\right) }[/math]); explicitly, this means that [math]\displaystyle{ \varnothing \neq N \cap S_{\geq i} }[/math] for every index [math]\displaystyle{ i \in I }[/math] and neighborhood [math]\displaystyle{ N }[/math] of [math]\displaystyle{ x. }[/math]

Every net of sets that is eventually empty converges to every point/subset. However, a net of sets converges to [math]\displaystyle{ \varnothing }[/math] if and only if it is eventually empty. No net of sets clusters at [math]\displaystyle{ \varnothing. }[/math] If a net of sets converges to [math]\displaystyle{ x }[/math] then it will cluster at [math]\displaystyle{ x }[/math] if and only if it is not eventually empty (which implies [math]\displaystyle{ x \neq \varnothing }[/math]). If [math]\displaystyle{ f : X \to Y }[/math] is a map and [math]\displaystyle{ y_\bull = \left(y_i\right)_{i \in I} }[/math] is a net (of points or of sets) then [math]\displaystyle{ f^{-1}\left(y_\bull\right) }[/math] converges to (respectively, clusters at) some given point or subset of [math]\displaystyle{ X }[/math] if and only if every neighborhood of it contains (respectively, intersects) some set of the form [math]\displaystyle{ f^{-1}\left(y_{\geq i}\right). }[/math] Moreover, the net [math]\displaystyle{ f^{-1}\left(y_\bull\right) }[/math] converges in [math]\displaystyle{ X }[/math] to some given point or subset if and only if this is true of [math]\displaystyle{ f^{-1}\left(\operatorname{Tails}\left(y_\bull\right)\right). }[/math]

Applications

Some applications are now given showing how nets of sets can be used to characterize various properties. In the statements below, unless indicated otherwise, [math]\displaystyle{ y }[/math] and the net [math]\displaystyle{ y_\bull }[/math] are points in [math]\displaystyle{ Y }[/math] (not sets) and the map [math]\displaystyle{ f : X \to Y }[/math] is not necessarily surjective.

• A map [math]\displaystyle{ f : X \to Y }[/math] is closed (meaning it sends closed sets to closed subset of [math]\displaystyle{ Y }[/math]) if and only if whenever [math]\displaystyle{ y_\bull \to y \text{ in } Y }[/math] then [math]\displaystyle{ f^{-1}\left(y_\bull\right) \to f^{-1}(y) \text{ in } X. }[/math]
  • In comparison, [math]\displaystyle{ f : X \to Y }[/math] is continuous if and only if whenever [math]\displaystyle{ x_\bull \to x \text{ in } X }[/math] then [math]\displaystyle{ f\left(x_\bull\right) \to f(x) \text{ in } Y. }[/math]
  This characterization remains true if [math]\displaystyle{ y_\bull \text{ and } y }[/math] are allowed to be sets (instead of restricted to being points) such that [math]\displaystyle{ y_\bull \to y \text{ in } Y. }[/math]

  Proof

  Assume [math]\displaystyle{ f : X \to Y }[/math] is closed and [math]\displaystyle{ y_\bull \to y \text{ in } Y. }[/math] If [math]\displaystyle{ y \not\in f(X) }[/math] then [math]\displaystyle{ y }[/math] is in the open set [math]\displaystyle{ Y \setminus f(X) }[/math] so that [math]\displaystyle{ y_\bull \to y \text{ in } Y }[/math] implies that [math]\displaystyle{ f^{-1}\left(y_\bull\right) }[/math] is eventually empty and thus that [math]\displaystyle{ f^{-1}\left(y_\bull\right) \to \varnothing = f^{-1}(y) }[/math] in [math]\displaystyle{ X. }[/math] So assume [math]\displaystyle{ f^{-1}(y) \neq \varnothing }[/math] and let [math]\displaystyle{ U }[/math] be an open neighborhood of [math]\displaystyle{ f^{-1}(y) }[/math] in [math]\displaystyle{ X. }[/math] It remains to show that [math]\displaystyle{ f^{-1}\left(y_{\geq i}\right) \subseteq U }[/math] for some index [math]\displaystyle{ i. }[/math] Since [math]\displaystyle{ f }[/math] is closed, [math]\displaystyle{ Y \setminus f(X \setminus U) }[/math] is an open neighborhood of [math]\displaystyle{ y }[/math] in [math]\displaystyle{ Y }[/math] so there must exists some index [math]\displaystyle{ i }[/math] such that [math]\displaystyle{ y_{\geq i} \subseteq Y \setminus f(X \setminus U). }[/math] This implies [math]\displaystyle{ f^{-1}\left(y_{\geq i}\right) \subseteq f^{-1}(Y \setminus f(X \setminus U)) }[/math] where the right hand side is a subset of [math]\displaystyle{ U, }[/math] as desired. For the converse, assume that [math]\displaystyle{ y_\bull \to y \text{ in } Y }[/math] implies [math]\displaystyle{ f^{-1}\left(y_\bull\right) \to f^{-1}(y) \text{ in } X. }[/math] Let [math]\displaystyle{ C \subseteq X }[/math] be closed and assume it is not empty. Let [math]\displaystyle{ y_\bull = \left(y_i\right)_{i \in I} }[/math] be a net in [math]\displaystyle{ f(C) }[/math] (meaning [math]\displaystyle{ y_i \in f(C) }[/math] for all [math]\displaystyle{ i }[/math]) and let [math]\displaystyle{ y \in Y }[/math] be such that [math]\displaystyle{ y_\bull \to y \text{ in } Y. }[/math] It remains to show that [math]\displaystyle{ y \in f(C). }[/math] The hypotheses guarantee that [math]\displaystyle{ f^{-1}\left(y_\bull\right) \to f^{-1}(y) \text{ in } X. }[/math] The fact that every fiber [math]\displaystyle{ f^{-1}\left(y_i\right) }[/math] is not empty and that these fibers converge to [math]\displaystyle{ f^{-1}(y) }[/math] imply that [math]\displaystyle{ f^{-1}(y) \neq \varnothing. }[/math] Since [math]\displaystyle{ X \setminus C }[/math] is open, were it true that [math]\displaystyle{ f^{-1}(y) \subseteq X \setminus C }[/math] then there would exist some index [math]\displaystyle{ i }[/math] such that [math]\displaystyle{ f^{-1}\left(y_{\geq i}\right) \subseteq X \setminus C, }[/math] which is impossible since [math]\displaystyle{ y_j \in f(C) }[/math] for every index [math]\displaystyle{ j. }[/math] Thus [math]\displaystyle{ f^{-1}(y) \not\subseteq X \setminus C }[/math] so there is some [math]\displaystyle{ c \in C \cap f^{-1}(y), }[/math] which proves that [math]\displaystyle{ y = f(c) \in f(C). }[/math] [math]\displaystyle{ \blacksquare }[/math]

• A map [math]\displaystyle{ f : X \to Y }[/math] is open (meaning it sends open sets to open subset of [math]\displaystyle{ Y }[/math]) if and only if whenever [math]\displaystyle{ x }[/math] is a point in [math]\displaystyle{ X }[/math] and [math]\displaystyle{ y_\bull }[/math] is a net that clusters at [math]\displaystyle{ f(x) \text{ in } Y, }[/math] then [math]\displaystyle{ f^{-1}\left(y_\bull\right) }[/math] clusters at [math]\displaystyle{ x \text{ in } X. }[/math]
  • In comparison, [math]\displaystyle{ f : X \to Y }[/math] is continuous if and only if whenever [math]\displaystyle{ x_\bull }[/math] is a net that clusters at a point [math]\displaystyle{ x \text{ in } X, }[/math] then [math]\displaystyle{ f\left(x_\bull\right) }[/math] clusters at [math]\displaystyle{ f(x) \text{ in } Y. }[/math]
  This characterization remains true if [math]\displaystyle{ y_\bull \text{ and } x }[/math] are allowed to be sets.

  Proof

  For the non-trivial direction, suppose that [math]\displaystyle{ f : X \to Y }[/math] is not an open map. Pick an open subset [math]\displaystyle{ U \subseteq X }[/math] such that [math]\displaystyle{ f(U) }[/math] is not open in [math]\displaystyle{ Y, }[/math] where non-openness means that there is some point [math]\displaystyle{ y \in f(U) }[/math] such that [math]\displaystyle{ f(U) }[/math] is not a neighborhood of [math]\displaystyle{ y }[/math] in [math]\displaystyle{ Y. }[/math] Explicitly, this means that [math]\displaystyle{ N \not\subseteq f(U) }[/math] for every neighborhood [math]\displaystyle{ N }[/math] of [math]\displaystyle{ y }[/math] in [math]\displaystyle{ Y, }[/math] which guarantees the existence of some [math]\displaystyle{ y_N \in N \setminus f(U). }[/math] Let [math]\displaystyle{ I := \mathcal{N}_Y(y) }[/math] denote the neighborhood filter of [math]\displaystyle{ y }[/math] in [math]\displaystyle{ Y }[/math] and direct it by [math]\displaystyle{ \,\supseteq\, }[/math] to make [math]\displaystyle{ y_\bull := \left(y_N\right)_{N \in I} }[/math] into a net that converges to [math]\displaystyle{ y }[/math] in [math]\displaystyle{ Y, }[/math] which implies that [math]\displaystyle{ y_\bull }[/math] clusters at [math]\displaystyle{ y }[/math] in [math]\displaystyle{ Y. }[/math] Because [math]\displaystyle{ y \in f(U), }[/math] there exists [math]\displaystyle{ x \in U \cap f^{-1}(y). }[/math] But [math]\displaystyle{ f^{-1}\left(y_\bull\right) }[/math] does not clusters at [math]\displaystyle{ x }[/math] since [math]\displaystyle{ f^{-1}\left(y_N\right) \cap U = \varnothing }[/math] for every [math]\displaystyle{ N \in I. }[/math] [math]\displaystyle{ \blacksquare }[/math] The alternative proof below is demonstrate how a prefilter can be used to construct a net of sets, which in turn can be used to construct a net of points. Because [math]\displaystyle{ f(U) }[/math] is not a neighborhood of [math]\displaystyle{ y, }[/math] the family [math]\displaystyle{ \mathcal{I} := \{N \setminus f(U) : N \in \mathcal{N}(y)\} }[/math] does not contain the empty set. If [math]\displaystyle{ N }[/math] and [math]\displaystyle{ N_2 }[/math] are neighborhood of [math]\displaystyle{ y }[/math] then the intersections [math]\displaystyle{ (N_2 \setminus f(U)) \cap (N \setminus f(U)) }[/math] and [math]\displaystyle{ N_2 \cap (N \setminus f(U)) }[/math] both equal [math]\displaystyle{ (N_2 \cap N) \setminus f(U), }[/math] which belongs to [math]\displaystyle{ \mathcal{I} }[/math] (since [math]\displaystyle{ N_2 \cap N \in \mathcal{N}(y) }[/math]) and is thus not empty. This shows that [math]\displaystyle{ \mathcal{I} }[/math] is a π-system and that it meshes with the neighborhood filter [math]\displaystyle{ \mathcal{N}(y). }[/math] In particular, [math]\displaystyle{ \mathcal{I} }[/math] is a prefilter that clusters at [math]\displaystyle{ y. }[/math] Moreover, [math]\displaystyle{ \mathcal{N}(y) \leq \mathcal{I} }[/math] because every [math]\displaystyle{ N \in \mathcal{N}(y) }[/math] contains [math]\displaystyle{ N \setminus f(U) \in \mathcal{I} }[/math] as a subset, which proves that [math]\displaystyle{ \mathcal{I} \to y. }[/math] Pick [math]\displaystyle{ x \in U \cap f^{-1}(y) }[/math] as before. The set [math]\displaystyle{ U }[/math] is thus a neighborhood of [math]\displaystyle{ x }[/math] that is disjoint from [math]\displaystyle{ f^{-1}(N \setminus f(U)) }[/math] for every neighborhood [math]\displaystyle{ N \in \mathcal{N}(y). }[/math] Thus [math]\displaystyle{ f^{-1}(\mathcal{I}) }[/math] does not cluster at [math]\displaystyle{ x }[/math] even though the prefilter [math]\displaystyle{ \mathcal{I} }[/math] clusters at [math]\displaystyle{ y. }[/math] Conclusion using nets of sets: Direct the above prefilter [math]\displaystyle{ \mathcal{I} }[/math] by [math]\displaystyle{ \,\supseteq\, }[/math] so that the identity map [math]\displaystyle{ \operatorname{id}_{\mathcal{I}} : (\mathcal{I}, \supseteq) \to \mathcal{I} }[/math] becomes a net of sets. This net clusters at (respectively, converges to) [math]\displaystyle{ y }[/math] because this is true of [math]\displaystyle{ \mathcal{I}. }[/math] But because [math]\displaystyle{ f^{-1}(\mathcal{I}) }[/math] does not cluster at [math]\displaystyle{ x, }[/math] neither does the net of preimages [math]\displaystyle{ (f^{-1}(I))_{I \in \mathcal{I}}. }[/math] Conclusion using nets of points: For every [math]\displaystyle{ I \in \mathcal{I}, }[/math] pick a point [math]\displaystyle{ y_I \in I. }[/math] Then [math]\displaystyle{ y_\bull := \left(y_I\right)_{I \in \mathcal{I}} }[/math] is a net that converges to [math]\displaystyle{ y }[/math] in [math]\displaystyle{ Y }[/math] (because this is true of the net of sets [math]\displaystyle{ \operatorname{id}_{\mathcal{I}} : (\mathcal{I}, \supseteq) \to \mathcal{I} }[/math]), which implies that [math]\displaystyle{ y_\bull }[/math] clusters at [math]\displaystyle{ y }[/math] in [math]\displaystyle{ Y. }[/math] But [math]\displaystyle{ f^{-1}\left(y_\bull\right) }[/math] does not clusters at [math]\displaystyle{ x }[/math] since [math]\displaystyle{ f^{-1}\left(y_I\right) \cap U \subseteq f^{-1}(I) \cap U = \varnothing }[/math] for every [math]\displaystyle{ I \in \mathcal{I}. }[/math] [math]\displaystyle{ \blacksquare }[/math]

• A map [math]\displaystyle{ f : X \to Y }[/math] is open if and only if whenever [math]\displaystyle{ y_\bull \to y \text{ in } Y }[/math] then any closed subset of [math]\displaystyle{ X }[/math] that contains [math]\displaystyle{ f^{-1}\left(y_\bull\right) }[/math] will necessarily also contain [math]\displaystyle{ f^{-1}(y). }[/math]
  • In comparison, by the closure characterization of continuity, [math]\displaystyle{ f : X \to Y }[/math] is continuous if and only if whenever [math]\displaystyle{ x_\bull \to x \text{ in } X }[/math] then any closed subset of [math]\displaystyle{ Y }[/math] that contains [math]\displaystyle{ f\left(x_\bull\right) }[/math] will necessarily also contain [math]\displaystyle{ f(x). }[/math]
  This characterization remains true if [math]\displaystyle{ y_\bull }[/math] is allowed to be a net of sets that is not eventually empty (instead of being a net of points) while [math]\displaystyle{ y \in Y }[/math] continues to be a point (such that [math]\displaystyle{ y_\bull \to y \text{ in } Y }[/math]); the same is true of the quotient map characterization below.

  Proof

  If [math]\displaystyle{ R \subseteq X }[/math] is any subset then it is readily verified that [math]\displaystyle{ f(X \setminus R) = Y \setminus \left\{y \in Y : f^{-1}(y) \subseteq R\right\}. }[/math] This implies that a map [math]\displaystyle{ f : X \to Y }[/math] is open if and only if whenever [math]\displaystyle{ C \subseteq X }[/math] is closed in [math]\displaystyle{ X }[/math] then [math]\displaystyle{ \left\{y \in Y ~:~ f^{-1}(y) \subseteq C\right\} }[/math] is closed in [math]\displaystyle{ Y. }[/math] This characterization of "open map" combined with the convergent net characterization of closed sets produces the desired conclusion: [math]\displaystyle{ f : X \to Y }[/math] is open if and only if whenever [math]\displaystyle{ y_\bull \to y \text{ in } Y }[/math] and [math]\displaystyle{ C \subseteq X }[/math] is a closed subset of [math]\displaystyle{ X }[/math] that contains [math]\displaystyle{ f^{-1}\left(y_\bull\right), }[/math] then necessarily [math]\displaystyle{ f^{-1}(y) \subseteq C. }[/math] [math]\displaystyle{ \blacksquare }[/math]

• A continuous surjection [math]\displaystyle{ f : X \to Y }[/math] is a quotient map if and only if whenever [math]\displaystyle{ y_\bull \to y \text{ in } Y }[/math] then any saturated closed subset of [math]\displaystyle{ X }[/math] that contains [math]\displaystyle{ f^{-1}\left(y_\bull\right) }[/math] will necessarily also contain [math]\displaystyle{ f^{-1}(y). }[/math] (A set [math]\displaystyle{ C \subseteq X }[/math] is saturated if [math]\displaystyle{ f^{-1}(f(C)) = C. }[/math])
• A subset [math]\displaystyle{ C \subseteq X }[/math] is closed in [math]\displaystyle{ X }[/math] if and only if for every point [math]\displaystyle{ x \in X }[/math] and every net [math]\displaystyle{ c_\bull }[/math] of subsets of [math]\displaystyle{ C }[/math] that is not eventually empty, if [math]\displaystyle{ c_\bull \to x \text{ in } X }[/math] then [math]\displaystyle{ x \in C. }[/math]
• A map [math]\displaystyle{ f : X \to Y }[/math] is continuous if and only if whenever [math]\displaystyle{ S }[/math] and [math]\displaystyle{ S_\bull }[/math] are sets or points in [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ S_\bull \to S \text{ in } X, }[/math] then [math]\displaystyle{ f\left(S_\bull\right) \to f(S) \text{ in } Y. }[/math]

  Proof

  The proof is essentially identical to the usual proof involving only nets of points. One direction (that whose conclusion is that [math]\displaystyle{ f }[/math] is continuous) only requires consideration of nets of points and so it is omitted. So suppose that the map is continuous and that [math]\displaystyle{ S_\bull \to S \text{ in } X. }[/math] Let [math]\displaystyle{ V \subseteq Y }[/math] be an open neighborhood of [math]\displaystyle{ f(S) }[/math] in [math]\displaystyle{ Y. }[/math] Then [math]\displaystyle{ f^{-1}(V) }[/math] is an open neighborhood of [math]\displaystyle{ S }[/math] in [math]\displaystyle{ X }[/math] so there exists some index [math]\displaystyle{ i }[/math] such that [math]\displaystyle{ S_{\geq i} \subseteq f^{-1}(V). }[/math] Thus [math]\displaystyle{ f\left(S_{\geq i}\right) \subseteq V, }[/math] as desired. [math]\displaystyle{ \blacksquare }[/math]

• A map [math]\displaystyle{ f : X \to Y }[/math] is continuous if and only if whenever [math]\displaystyle{ S_\bull }[/math] is a net of sets or points in [math]\displaystyle{ X }[/math] that clusters at (respectively, converges to) some given point or subset [math]\displaystyle{ S }[/math] of [math]\displaystyle{ X, }[/math] then [math]\displaystyle{ f\left(S_\bull\right) := \left(f\left(S_i\right)\right)_{i \in I} }[/math] clusters at (respectively, converges to) [math]\displaystyle{ f(S) }[/math] in [math]\displaystyle{ Y. }[/math]

Topologizing the set of prefilters

Starting with nothing more than a set [math]\displaystyle{ X, }[/math] it is possible to topologize the set [math]\displaystyle{ \mathbb{P} := \operatorname{Prefilters}(X) }[/math] of all filter bases on [math]\displaystyle{ X }[/math] with the Stone topology, which is named after Marshall Harvey Stone.

To reduce confusion, this article will adhere to the following notational conventions:

• Lower case letters for elements [math]\displaystyle{ x \in X. }[/math]
• Upper case letters for subsets [math]\displaystyle{ S \subseteq X. }[/math]
• Upper case calligraphy letters for subsets [math]\displaystyle{ \mathcal{B} \subseteq \wp(X) }[/math] (or equivalently, for elements [math]\displaystyle{ \mathcal{B} \in \wp(\wp(X)), }[/math] such as prefilters).
• Upper case double–struck letters for subsets [math]\displaystyle{ \mathbb{P} \subseteq \wp(\wp(X)). }[/math]

For every [math]\displaystyle{ S \subseteq X, }[/math] let [math]\displaystyle{ \mathbb{O}(S) := \left\{\mathcal{B} \in \mathbb{P} ~:~ S \in \mathcal{B}^{\uparrow X}\right\} }[/math] where [math]\displaystyle{ \mathbb{O}(X) = \mathbb{P} \text{ and } \mathbb{O}(\varnothing) = \varnothing. }[/math] These sets will be the basic open subsets of the Stone topology. If [math]\displaystyle{ R \subseteq S \subseteq X }[/math] then [math]\displaystyle{ \left\{\mathcal{B} \in \wp(\wp(X)) ~:~ R \in \mathcal{B}^{\uparrow X}\right\} ~\subseteq~ \left\{\mathcal{B} \in \wp(\wp(X)) ~:~ S \in \mathcal{B}^{\uparrow X}\right\}. }[/math]

From this inclusion, it is possible to deduce all of the subset inclusions displayed below with the exception of [math]\displaystyle{ \mathbb{O}(R \cap S) ~\supseteq~ \mathbb{O}(R) \cap \mathbb{O}(S). }[/math] For all [math]\displaystyle{ R \subseteq S \subseteq X, }[/math] [math]\displaystyle{ \mathbb{O}(R \cap S) ~=~ \mathbb{O}(R) \cap \mathbb{O}(S) ~\subseteq~ \mathbb{O}(R) \cup \mathbb{O}(S) ~\subseteq~ \mathbb{O}(R \cup S) }[/math] where in particular, the equality [math]\displaystyle{ \mathbb{O}(R \cap S) = \mathbb{O}(R) \cap \mathbb{O}(S) }[/math] shows that the family [math]\displaystyle{ \{\mathbb{O}(S) ~:~ S \subseteq X\} }[/math] is a [math]\displaystyle{ \pi }[/math]–system that forms a basis for a topology on [math]\displaystyle{ \mathbb{P} }[/math] called the Stone topology. It is henceforth assumed that [math]\displaystyle{ \mathbb{P} }[/math] carries this topology and that any subset of [math]\displaystyle{ \mathbb{P} }[/math] carries the induced subspace topology.

In contrast to most other general constructions of topologies (for example, the product, quotient, subspace topologies, etc.), this topology on [math]\displaystyle{ \mathbb{P} }[/math] was defined without using anything other than the set [math]\displaystyle{ X; }[/math] there were no preexisting structures or assumptions on [math]\displaystyle{ X }[/math] so this topology is completely independent of everything other than [math]\displaystyle{ X }[/math] (and its subsets).

The following criteria can be used for checking for points of closure and neighborhoods. If [math]\displaystyle{ \mathbb{B} \subseteq \mathbb{P} \text{ and } \mathcal{F} \in \mathbb{P} }[/math] then:

• Closure in [math]\displaystyle{ \mathbb{P} }[/math]: [math]\displaystyle{ \ \mathcal{F} }[/math] belongs to the closure of [math]\displaystyle{ \mathbb{B} \text{ in } \mathbb{P} }[/math] if and only if [math]\displaystyle{ \mathcal{F} \subseteq {\textstyle\bigcup\limits_{\mathcal{B} \in \mathbb{B}}} \mathcal{B}^{\uparrow X}. }[/math]
• Neighborhoods in [math]\displaystyle{ \mathbb{P} }[/math]: [math]\displaystyle{ \ \mathbb{B} }[/math] is a neighborhood of [math]\displaystyle{ \mathcal{F} \text{ in } \mathbb{P} }[/math] if and only if there exists some [math]\displaystyle{ F \in \mathcal{F} }[/math] such that [math]\displaystyle{ \mathbb{O}(F) = \left\{\mathcal{B} \in \mathbb{P} ~:~ F \in \mathcal{B}^{\uparrow X}\right\} \subseteq \mathbb{B} }[/math] (that is, such that for all [math]\displaystyle{ \mathcal{B} \in \mathbb{P}, \text{ if } F \in \mathcal{B}^{\uparrow X} \text{ then } \mathcal{B} \in \mathbb{B} }[/math]).

It will be henceforth assumed that [math]\displaystyle{ X \neq \varnothing }[/math] because otherwise [math]\displaystyle{ \mathbb{P} = \varnothing }[/math] and the topology is [math]\displaystyle{ \{\varnothing\}, }[/math] which is uninteresting.

Subspace of ultrafilters

The set of ultrafilters on [math]\displaystyle{ X }[/math] (with the subspace topology) is a Stone space, meaning that it is compact, Hausdorff, and totally disconnected. If [math]\displaystyle{ X }[/math] has the discrete topology then the map [math]\displaystyle{ \beta : X \to \operatorname{UltraFilters}(X), }[/math] defined by sending [math]\displaystyle{ x \in X }[/math] to the principal ultrafilter at [math]\displaystyle{ x, }[/math] is a topological embedding whose image is a dense subset of [math]\displaystyle{ \operatorname{UltraFilters}(X) }[/math] (see the article Stone–Čech compactification for more details).

Relationships between topologies on [math]\displaystyle{ X }[/math] and the Stone topology on [math]\displaystyle{ \mathbb{P} }[/math]

Every [math]\displaystyle{ \tau \in \operatorname{Top}(X) }[/math] induces a canonical map [math]\displaystyle{ \mathcal{N}_\tau : X \to \operatorname{Filters}(X) }[/math] defined by [math]\displaystyle{ x \mapsto \mathcal{N}_\tau(x), }[/math] which sends [math]\displaystyle{ x \in X }[/math] to the neighborhood filter of [math]\displaystyle{ x \text{ in } (X, \tau). }[/math] If [math]\displaystyle{ \tau, \sigma \in \operatorname{Top}(X) }[/math] then [math]\displaystyle{ \tau = \sigma }[/math] if and only if [math]\displaystyle{ \mathcal{N}_\tau = \mathcal{N}_\sigma. }[/math] Thus every topology [math]\displaystyle{ \tau \in \operatorname{Top}(X) }[/math] can be identified with the canonical map [math]\displaystyle{ \mathcal{N}_\tau \in \operatorname{Func}(X; \mathbb{P}), }[/math] which allows [math]\displaystyle{ \operatorname{Top}(X) }[/math] to be canonically identified as a subset of [math]\displaystyle{ \operatorname{Func}(X; \mathbb{P}) }[/math] (as a side note, it is now possible to place on [math]\displaystyle{ \operatorname{Func}(X; \mathbb{P}), }[/math] and thus also on [math]\displaystyle{ \operatorname{Top}(X), }[/math] the topology of pointwise convergence on [math]\displaystyle{ X }[/math] so that it now makes sense to talk about things such as sequences of topologies on [math]\displaystyle{ X }[/math] converging pointwise). For every [math]\displaystyle{ \tau \in \operatorname{Top}(X), }[/math] the surjection [math]\displaystyle{ \mathcal{N}_\tau : (X, \tau) \to \operatorname{image} \mathcal{N}_\tau }[/math] is always continuous, closed, and open, but it is injective if and only if [math]\displaystyle{ \tau \text{ is } T_0 }[/math] (that is, a Kolmogorov space). In particular, for every [math]\displaystyle{ T_0 }[/math] topology [math]\displaystyle{ \tau \text{ on } X, }[/math] the map [math]\displaystyle{ \mathcal{N}_\tau : (X, \tau) \to \mathbb{P} }[/math] is a topological embedding (said differently, every Kolmogorov space is a topological subspace of the space of prefilters).

In addition, if [math]\displaystyle{ \mathfrak{F} : X \to \operatorname{Filters}(X) }[/math] is a map such that [math]\displaystyle{ x \in \ker \mathfrak{F}(x) := {\textstyle\bigcap\limits_{F \in \mathfrak{F}(x)}} F \text{ for every } x \in X }[/math] (which is true of [math]\displaystyle{ \mathfrak{F} := \mathcal{N}_\tau, }[/math] for instance), then for every [math]\displaystyle{ x \in X \text{ and } F \in \mathfrak{F}(x), }[/math] the set [math]\displaystyle{ \mathfrak{F}(F) = \{\mathfrak{F}(f) : f \in F\} }[/math] is a neighborhood (in the subspace topology) of [math]\displaystyle{ \mathfrak{F}(x) \text{ in } \operatorname{image} \mathfrak{F}. }[/math]

See also

• Characterizations of the category of topological spaces
• Convergence space – Generalization of the notion of convergence that is found in general topology
• Filtration (mathematics)
• Filtration (probability theory) – Model of information available at a given point of a random process
• Fréchet filter
• Generic filter
• Ideal (set theory) – Non-empty family of sets that is closed under finite unions and subsets

Notes

Proofs

References

• Template:Adams Franzosa Introduction to Topology Pure and Applied
• Template:Arkhangel'skii Ponomarev Fundamentals of General Topology Problems and Exercises
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. 
• Bourbaki, Nicolas (1989). General Topology: Chapters 1–4. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129. https://doku.pub/documents/31425779-nicolas-bourbaki-general-topology-part-i1pdf-30j71z37920w. 
• Template:Bourbaki General Topology Part II Chapters 5-10
• Bourbaki, Nicolas (1987). Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. 2. Berlin New York: Springer-Verlag. ISBN 978-3-540-42338-6. OCLC 17499190. http://www.numdam.org/item?id=AIF_1950__2__5_0. 
• Burris, Stanley; Sankappanavar, Hanamantagouda P. (2012). A Course in Universal Algebra. Springer-Verlag. ISBN 978-0-9880552-0-9. http://www.thoralf.uwaterloo.ca/htdocs/ualg.html. 
• Cartan, Henri (1937a). "Théorie des filtres". Comptes rendus hebdomadaires des séances de l'Académie des sciences 205: 595–598. http://gallica.bnf.fr/ark:/12148/bpt6k3157c/f594.image. 
• Cartan, Henri (1937b). "Filtres et ultrafiltres". Comptes rendus hebdomadaires des séances de l'Académie des sciences 205: 777–779. http://gallica.bnf.fr/ark:/12148/bpt6k3157c/f776.image. 
• Template:Comfort Negrepontis The Theory of Ultrafilters 1974
• Template:Császár General Topology
• Template:Dixmier General Topology
• Dolecki, Szymon; Mynard, Frederic (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917. 
• Dugundji, James (1966). Topology. Boston: Allyn and Bacon. ISBN 978-0-697-06889-7. OCLC 395340485. 
• Template:Dunford Schwartz Linear Operators Part 1 General Theory
• Edwards, Robert E. (Jan 1, 1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138. 
• Template:Howes Modern Analysis and Topology 1995
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. 
• Jech, Thomas (2006). Set Theory: The Third Millennium Edition, Revised and Expanded. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-44085-7. OCLC 50422939. 
• Template:Joshi Introduction to General Topology
• Template:Kelley General Topology
• Köthe, Gottfried (1969). Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. 159. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. OCLC 840293704. 
• MacIver R., David (1 July 2004). "Filters in Analysis and Topology". http://www.efnet-math.org/~david/mathematics/filters.pdf.  (Provides an introductory review of filters in topology and in metric spaces.)
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. 
• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250. 
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. 
• Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365. 
• Template:Schubert Topology
• Trèves, François (August 6, 2006). Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. 
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. 
• Template:Wilansky Topology for Analysis 2008
• Willard, Stephen (2004). General Topology. Dover Books on Mathematics (First ed.). Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.